GCSE Maths: Highest common factor and lowest common multiple (Higher)
Part of the GCSE Maths factors, multiples and primes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Highest common factor and lowest common multiple
In Section 2.1 you met common factors and common multiples. Two special ones are used a lot.
The highest common factor (HCF) of two numbers is the biggest number that divides into both. We use it to cut things into the largest equal pieces, or to simplify fractions (Chapter 3).
The lowest common multiple (LCM) is the smallest number that is in both times tables. We use it when two things repeat and we want to know when they line up, or to find a common denominator.
Method 1: listing. For the HCF, list the factors of each number and find the biggest number in both lists. For the LCM, list the multiples of each number until you find the first one in both lists.
Method 2: prime factors and a Venn diagram. This method works for big numbers where lists would be long. Write each number as a product of primes. Draw two overlapping circles, one for each number. Put the primes that both numbers share in the middle, and the leftover primes in the outer parts.
HCF = multiply the numbers in the middle. LCM = multiply all the numbers in the diagram (the middle ones only once). Why? The middle holds what both numbers have in common, which builds the biggest shared factor. The LCM has to contain everything that either number needs, and using the shared primes only once keeps it as small as possible.
Which one do I need? If the answer must be smaller than or equal to both numbers (cutting, sharing, grouping), it is the HCF. If it is bigger than or equal to both (meeting again, repeating, buying enough), it is the LCM.
Key ideas
- HCF = the biggest number that divides into both. LCM = the smallest number in both times tables.
- Listing works for small numbers. For larger numbers use prime factors and a Venn diagram.
- Venn diagram: HCF = multiply the middle; LCM = multiply everything in the diagram.
- HCF ≤ both numbers; LCM ≥ both numbers.
- Check: HCF × LCM = the two numbers multiplied together.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 24 and 36. Here we list the factors.
- List the factors of 24. These are all the numbers that divide into it exactly.Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- List the factors of 36. Do the same for the other number.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 2, 3, 4, 6, 12
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 12
Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 6 times table and the 8 times table. Here we list the multiples.
- List the multiples of 6. Count up in 6s. Keep going until you have passed a number that is also in the other list.Multiples of 6: 6, 12, 18, 24, ...
- List the multiples of 8. Count up in 8s in the same way.Multiples of 8: 8, 16, 24, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 24
Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 60 and 84.
- Write 60 and 84 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 2 × 3 = 12
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 5 × 2 × 2 × 3 × 7 = 420
Check. HCF × LCM = 12 × 420 = 5040 and 60 × 84 = 5040. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 12 times table and the 18 times table. Here we use prime factors in a Venn diagram.
- Decide what the question is asking. The buses leave together again after a number of minutes that is in both the 12 times table and the 18 times table. We want the first time, so we need the lowest common multiple.Find the lowest common multiple of 12 and 18.
- Write 12 and 18 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.12 = 2 × 2 × 3
18 = 2 × 3 × 3 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 3 × 3 = 36
- Answer the question in words. Always finish by saying what the number means.They leave together again after 36 minutes, which is 9:36 am.
Check. HCF × LCM = 6 × 36 = 216 and 12 × 18 = 216. They are equal. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write each number as a product of prime factors in index form. Use a factor tree or repeated division by primes.24 = 23 × 3
36 = 22 × 32
60 = 22 × 3 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: 24: 23 | 36: 22 | 60: 22
Prime 3: 24: 31 | 36: 32 | 60: 31
Prime 5: 24: 50 | 36: 50 | 60: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 2, so 22
3: smallest power is 1, so 3
5: smallest power is 0, so it is left out
HCF = 22 × 3 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 3, so 23
3: largest power is 2, so 32
5: largest power is 1, so 5
LCM = 23 × 32 × 5
Check. The HCF divides every number: 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5. Every number divides the LCM: 360 ÷ 24 = 15, 360 ÷ 36 = 10, 360 ÷ 60 = 6. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write down the numbers in index form. They are already products of primes.A = 23 × 3 × 52
B = 22 × 32 × 7 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: A: 23 | B: 22
Prime 3: A: 31 | B: 32
Prime 5: A: 52 | B: 50
Prime 7: A: 70 | B: 71 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 2, so 22
3: smallest power is 1, so 3
5: smallest power is 0, so it is left out
7: smallest power is 0, so it is left out
HCF = 22 × 3 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 3, so 23
3: largest power is 2, so 32
5: largest power is 2, so 52
7: largest power is 1, so 7
LCM = 23 × 32 × 52 × 7
Check. The HCF divides every number: 600 ÷ 12 = 50, 252 ÷ 12 = 21. Every number divides the LCM: 12,600 ÷ 600 = 21, 12,600 ÷ 252 = 50. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Use the fact that HCF × LCM = the product of the two numbers.
- Write the fact. For any two numbers a and b, HCF × LCM = a × b. (In a Venn diagram, the HCF and LCM together contain every prime of both numbers.)HCF × LCM = x × 18
- Substitute the known values. 6 × 90 = x × 18
540 = 18x - Solve. Divide both sides by 18.x = 540 ÷ 18 = 30
Check. HCF(30, 18): 30 = 2 × 3 × 5 and 18 = 2 × 3 × 3 share 2 × 3 = 6. LCM = 2 × 3 × 5 × 3 = 90. ✓
Watch out. This shortcut only works for two numbers.
Watch out
Do not mix the two up: HCF is a factor (small), LCM is a multiple (big). And the LCM is not always the two numbers multiplied together: LCM of 6 and 8 is 24, not 48.Exercise 2D HCF and LCM
Non-calculatorKey words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 12 and 18. Here we list the factors.
- List the factors of 12. These are all the numbers that divide into it exactly.Factors of 12: 1, 2, 3, 4, 6, 12
- List the factors of 18. Do the same for the other number.Factors of 18: 1, 2, 3, 6, 9, 18
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 2, 3, 6
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 6
Check. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 8 and 20. Here we list the factors.
- List the factors of 8. These are all the numbers that divide into it exactly.Factors of 8: 1, 2, 4, 8
- List the factors of 20. Do the same for the other number.Factors of 20: 1, 2, 4, 5, 10, 20
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 2, 4
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 4
Check. 8 ÷ 4 = 2 and 20 ÷ 4 = 5, and 2 and 5 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 15 and 25. Here we list the factors.
- List the factors of 15. These are all the numbers that divide into it exactly.Factors of 15: 1, 3, 5, 15
- List the factors of 25. Do the same for the other number.Factors of 25: 1, 5, 25
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 5
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 5
Check. 15 ÷ 5 = 3 and 25 ÷ 5 = 5, and 3 and 5 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 16 and 40. Here we list the factors.
- List the factors of 16. These are all the numbers that divide into it exactly.Factors of 16: 1, 2, 4, 8, 16
- List the factors of 40. Do the same for the other number.Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 2, 4, 8
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 8
Check. 16 ÷ 8 = 2 and 40 ÷ 8 = 5, and 2 and 5 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 14 and 21. Here we list the factors.
- List the factors of 14. These are all the numbers that divide into it exactly.Factors of 14: 1, 2, 7, 14
- List the factors of 21. Do the same for the other number.Factors of 21: 1, 3, 7, 21
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 7
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 7
Check. 14 ÷ 7 = 2 and 21 ÷ 7 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 24 and 36. Here we list the factors.
- List the factors of 24. These are all the numbers that divide into it exactly.Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- List the factors of 36. Do the same for the other number.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1, 2, 3, 4, 6, 12
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 12
Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 3 times table and the 4 times table. Here we list the multiples.
- List the multiples of 3. Count up in 3s. Keep going until you have passed a number that is also in the other list.Multiples of 3: 3, 6, 9, 12, ...
- List the multiples of 4. Count up in 4s in the same way.Multiples of 4: 4, 8, 12, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 12
Check. 12 ÷ 3 = 4 and 12 ÷ 4 = 3, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 6 times table and the 8 times table. Here we list the multiples.
- List the multiples of 6. Count up in 6s. Keep going until you have passed a number that is also in the other list.Multiples of 6: 6, 12, 18, 24, ...
- List the multiples of 8. Count up in 8s in the same way.Multiples of 8: 8, 16, 24, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 24
Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 10 times table and the 15 times table. Here we list the multiples.
- List the multiples of 10. Count up in 10s. Keep going until you have passed a number that is also in the other list.Multiples of 10: 10, 20, 30, ...
- List the multiples of 15. Count up in 15s in the same way.Multiples of 15: 15, 30, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 30
Check. 30 ÷ 10 = 3 and 30 ÷ 15 = 2, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 4 times table and the 18 times table. Here we list the multiples.
- List the multiples of 4. Count up in 4s. Keep going until you have passed a number that is also in the other list.Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- List the multiples of 18. Count up in 18s in the same way.Multiples of 18: 18, 36, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 36
Check. 36 ÷ 4 = 9 and 36 ÷ 18 = 2, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 9 times table and the 12 times table. Here we list the multiples.
- List the multiples of 9. Count up in 9s. Keep going until you have passed a number that is also in the other list.Multiples of 9: 9, 18, 27, 36, ...
- List the multiples of 12. Count up in 12s in the same way.Multiples of 12: 12, 24, 36, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 36
Check. 36 ÷ 9 = 4 and 36 ÷ 12 = 3, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 5 times table and the 7 times table. Here we list the multiples.
- List the multiples of 5. Count up in 5s. Keep going until you have passed a number that is also in the other list.Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...
- List the multiples of 7. Count up in 7s in the same way.Multiples of 7: 7, 14, 21, 28, 35, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 35
Check. 35 ÷ 5 = 7 and 35 ÷ 7 = 5, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 18 and 24.
- Write 18 and 24 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 = 6
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 3 × 2 × 3 × 2 × 2 = 72
Check. HCF × LCM = 6 × 72 = 432 and 18 × 24 = 432. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 30 and 42.
- Write 30 and 42 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.30 = 2 × 3 × 5
42 = 2 × 3 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 = 6
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 5 × 2 × 3 × 7 = 210
Check. HCF × LCM = 6 × 210 = 1260 and 30 × 42 = 1260. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 36 and 48.
- Write 36 and 48 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 2 × 3 = 12
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 3 × 2 × 2 × 3 × 2 × 2 = 144
Check. HCF × LCM = 12 × 144 = 1728 and 36 × 48 = 1728. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 40 and 56.
- Write 40 and 56 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.40 = 2 × 2 × 2 × 5
56 = 2 × 2 × 2 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 2 × 2 = 8
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 5 × 2 × 2 × 2 × 7 = 280
Check. HCF × LCM = 8 × 280 = 2240 and 40 × 56 = 2240. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 45 and 75.
- Write 45 and 75 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.45 = 3 × 3 × 5
75 = 3 × 5 × 5 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 3 × 5 = 15
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 3 × 3 × 5 × 5 = 225
Check. HCF × LCM = 15 × 225 = 3375 and 45 × 75 = 3375. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 60 and 90.
- Write 60 and 90 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.60 = 2 × 2 × 3 × 5
90 = 2 × 3 × 3 × 5 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 × 5 = 30
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 3 × 5 × 3 = 180
Check. HCF × LCM = 30 × 180 = 5400 and 60 × 90 = 5400. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 84 and 126.
- Write 84 and 126 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 × 7 = 42
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 3 × 7 × 3 = 252
Check. HCF × LCM = 42 × 252 = 10584 and 84 × 126 = 10584. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 72 and 108.
- Write 72 and 108 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.72 = 2 × 2 × 2 × 3 × 3
108 = 2 × 2 × 3 × 3 × 3 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 2 × 3 × 3 = 36
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 2 × 3 × 3 × 3 = 216
Check. HCF × LCM = 36 × 216 = 7776 and 72 × 108 = 7776. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 120 and 150.
- Write 120 and 150 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.120 = 2 × 2 × 2 × 3 × 5
150 = 2 × 3 × 5 × 5 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 × 5 = 30
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 2 × 3 × 5 × 5 = 600
Check. HCF × LCM = 30 × 600 = 18000 and 120 × 150 = 18000. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 168 and 252.
- Write 168 and 252 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.168 = 2 × 2 × 2 × 3 × 7
252 = 2 × 2 × 3 × 3 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 2 × 3 × 7 = 84
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 2 × 2 × 2 × 3 × 7 × 3 = 504
Check. HCF × LCM = 84 × 504 = 42336 and 168 × 252 = 42336. They are equal. ✓
Watch out. Check you give both answers, labelled HCF and LCM.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 15 times table and the 20 times table. Here we use prime factors in a Venn diagram.
- Decide what the question is asking. The buses are together again after a number of minutes in both the 15 and 20 times tables. We want the first time, so the lowest common multiple.Find the lowest common multiple of 15 and 20.
- Write 15 and 20 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.15 = 3 × 5
20 = 2 × 2 × 5 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.LCM = 3 × 5 × 2 × 2 = 60
- Answer the question in words. Always finish by saying what the number means.They leave together again after 60 minutes, which is 9:00 am.
Check. HCF × LCM = 5 × 60 = 300 and 15 × 20 = 300. They are equal. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 84 and 126. Here we use prime factors in a Venn diagram.
- Decide what the question is asking. The piece length must divide into 84 and into 126 exactly, and we want it as long as possible. That is the highest common factor.Find the highest common factor of 84 and 126.
- Write 84 and 126 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7 - Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
- HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.HCF = 2 × 3 × 7 = 42
- Answer the question in words. Always finish by saying what the number means.The longest possible piece is 42 cm (2 pieces from the 84 cm ribbon and 3 from the 126 cm ribbon).
Check. HCF × LCM = 42 × 252 = 10584 and 84 × 126 = 10584. They are equal. ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The number of pencils is a multiple of 12 and the number of rubbers is a multiple of 15. We want them to be equal, and as small as possible: the LCM.
- Find the LCM of 12 and 15. List the multiples. 12: 12, 24, 36, 48, 60. 15: 15, 30, 45, 60.LCM = 60
- Find the number of pencil packs. Divide 60 by the pack size.60 ÷ 12 = 5 packs
- Find the number of rubber packs. 60 ÷ 15 = 4 packs
- Answer. Say what each number means.5 packs of pencils and 4 packs of rubbers, which is 60 of each.
Check. 5 × 12 = 60 and 4 × 15 = 60. ✓
Watch out. The question asks for packs, not for the number of items.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. They flash together every LCM seconds. Then see how many of those fit into 5 minutes.
- Find the LCM of 20 and 30. List multiples. 20: 20, 40, 60. 30: 30, 60.LCM = 60 seconds
- Change 5 minutes to seconds. 1 minute = 60 seconds.5 × 60 = 300 seconds
- Count how many lots of 60 seconds fit in 300 seconds. They flash together after 60, 120, 180, 240 and 300 seconds.300 ÷ 60 = 5
Check. Times together: 60, 120, 180, 240, 300 seconds. That is 5. ✓
Watch out. Do not count the flash at 8:00 pm: the question says “more” times.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The number of bags must divide into 48 and 72, and we want the most bags: the HCF.
- Find the HCF of 48 and 72. 48 = 24 × 3 and 72 = 23 × 32. Smallest powers: 23 and 3.HCF = 23 × 3 = 8 × 3 = 24
- Find how many of each biscuit go in a bag. Share each type equally between the 24 bags.48 ÷ 24 = 2 chocolate
72 ÷ 24 = 3 oat - Answer. 24 bags, each with 2 chocolate and 3 oat biscuits.
Check. 24 × 2 = 48 and 24 × 3 = 72. ✓
Method. A is the product of the primes in the left circle, including the middle.
- Find the primes in circle A. Circle A contains the left part and the middle.2, 2, 3, 5
- Multiply them. Multiply in stages.2 × 2 = 4, 4 × 3 = 12, 12 × 5 = 60
Check. 60 = 22 × 3 × 5. ✓
Method. B is the product of the primes in the right circle, including the middle.
- Find the primes in circle B. Circle B contains the middle and the right part.3, 5, 7
- Multiply them. 3 × 5 = 15, 15 × 7 = 105
Check. 105 = 3 × 5 × 7. ✓
Method. The HCF is the product of the numbers in the middle.
- Find the numbers in the middle. These are shared by both.3 and 5
- Multiply. 3 × 5 = 15
Check. 15 divides 60 (60 ÷ 15 = 4) and 105 (105 ÷ 15 = 7). ✓
Method. The LCM is the product of all the numbers in the diagram.
- Write all the numbers in the diagram. Count the middle ones once.2, 2, 3, 5, 7
- Multiply. 2 × 2 × 3 × 5 = 60 and 60 × 7 = 420
Check. 420 ÷ 60 = 7 and 420 ÷ 105 = 4, both whole numbers. ✓
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Look for two numbers where one is a multiple of the other.
- Try 4 and 8. 8 is in the 4 times table.Multiples of 4: 4, 8, 12 ... Multiples of 8: 8, 16 ...
- Find the LCM. The first number in both lists.LCM = 8
- Compare. The LCM is not bigger than 8; it is equal to it.Sara is wrong. The LCM is at least as big as the larger number, but can be equal to it.
Check. 8 ÷ 4 = 2 and 8 ÷ 8 = 1. ✓
Watch out. One counter-example is enough.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The highest common factor (HCF) is the biggest number that divides into both 8 and 15. Here we list the factors.
- List the factors of 8. These are all the numbers that divide into it exactly.Factors of 8: 1, 2, 4, 8
- List the factors of 15. Do the same for the other number.Factors of 15: 1, 3, 5, 15
- Find the common factors. These are the numbers that appear in both lists.Common factors: 1
- Pick the highest one. The highest common factor is the biggest number in the common list.HCF = 1
Check. The only common factor is 1, so the HCF is 1 (the numbers are called coprime). ✓
Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The lowest common multiple (LCM) is the smallest number that is in both the 8 times table and the 15 times table. Here we list the multiples.
- List the multiples of 8. Count up in 8s. Keep going until you have passed a number that is also in the other list.Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, ...
- List the multiples of 15. Count up in 15s in the same way.Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, ...
- Find the first number that is in both lists. That is the lowest common multiple.LCM = 120
Check. 120 ÷ 8 = 15 and 120 ÷ 15 = 8, both whole numbers. And no smaller number appears in both lists. ✓
Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Multiply the numbers and compare with the LCM.
- Multiply 8 and 15. 8 × 15 = 120
- Compare. This is equal to the LCM from part (b).LCM = 120 = 8 × 15
- Explain. The numbers share no prime factor (HCF = 1), so nothing is counted twice in the Venn diagram.When the HCF is 1, the LCM is the product of the two numbers.
Check. HCF × LCM = 1 × 120 = 120 = 8 × 15. ✓
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write each number as a product of prime factors in index form. Use a factor tree or repeated division by primes.12 = 22 × 3
18 = 2 × 32
30 = 2 × 3 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: 12: 22 | 18: 21 | 30: 21
Prime 3: 12: 31 | 18: 32 | 30: 31
Prime 5: 12: 50 | 18: 50 | 30: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 1, so 2
3: smallest power is 1, so 3
5: smallest power is 0, so it is left out
HCF = 2 × 3 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 2, so 22
3: largest power is 2, so 32
5: largest power is 1, so 5
LCM = 22 × 32 × 5
Check. The HCF divides every number: 12 ÷ 6 = 2, 18 ÷ 6 = 3, 30 ÷ 6 = 5. Every number divides the LCM: 180 ÷ 12 = 15, 180 ÷ 18 = 10, 180 ÷ 30 = 6. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write each number as a product of prime factors in index form. Use a factor tree or repeated division by primes.24 = 23 × 3
36 = 22 × 32
60 = 22 × 3 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: 24: 23 | 36: 22 | 60: 22
Prime 3: 24: 31 | 36: 32 | 60: 31
Prime 5: 24: 50 | 36: 50 | 60: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 2, so 22
3: smallest power is 1, so 3
5: smallest power is 0, so it is left out
HCF = 22 × 3 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 3, so 23
3: largest power is 2, so 32
5: largest power is 1, so 5
LCM = 23 × 32 × 5
Check. The HCF divides every number: 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5. Every number divides the LCM: 360 ÷ 24 = 15, 360 ÷ 36 = 10, 360 ÷ 60 = 6. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write each number as a product of prime factors in index form. Use a factor tree or repeated division by primes.20 = 22 × 5
30 = 2 × 3 × 5
45 = 32 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: 20: 22 | 30: 21 | 45: 20
Prime 3: 20: 30 | 30: 31 | 45: 32
Prime 5: 20: 51 | 30: 51 | 45: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 0, so it is left out
3: smallest power is 0, so it is left out
5: smallest power is 1, so 5
HCF = 5 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 2, so 22
3: largest power is 2, so 32
5: largest power is 1, so 5
LCM = 22 × 32 × 5
Check. The HCF divides every number: 20 ÷ 5 = 4, 30 ÷ 5 = 6, 45 ÷ 5 = 9. Every number divides the LCM: 180 ÷ 20 = 9, 180 ÷ 30 = 6, 180 ÷ 45 = 4. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write each number as a product of prime factors in index form. Use a factor tree or repeated division by primes.16 = 24
24 = 23 × 3
40 = 23 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: 16: 24 | 24: 23 | 40: 23
Prime 3: 16: 30 | 24: 31 | 40: 30
Prime 5: 16: 50 | 24: 50 | 40: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 3, so 23
3: smallest power is 0, so it is left out
5: smallest power is 0, so it is left out
HCF = 23 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 4, so 24
3: largest power is 1, so 3
5: largest power is 1, so 5
LCM = 24 × 3 × 5
Check. The HCF divides every number: 16 ÷ 8 = 2, 24 ÷ 8 = 3, 40 ÷ 8 = 5. Every number divides the LCM: 240 ÷ 16 = 15, 240 ÷ 24 = 10, 240 ÷ 40 = 6. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write down the numbers in index form. They are already products of primes.A = 23 × 3 × 52
B = 22 × 32 × 7 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: A: 23 | B: 22
Prime 3: A: 31 | B: 32
Prime 5: A: 52 | B: 50
Prime 7: A: 70 | B: 71 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 2, so 22
3: smallest power is 1, so 3
5: smallest power is 0, so it is left out
7: smallest power is 0, so it is left out
HCF = 22 × 3 = 12 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 3, so 23
3: largest power is 2, so 32
5: largest power is 2, so 52
7: largest power is 1, so 7
LCM = 23 × 32 × 52 × 7 = 12,600
Check. The HCF divides every number: 600 ÷ 12 = 50, 252 ÷ 12 = 21. Every number divides the LCM: 12,600 ÷ 600 = 21, 12,600 ÷ 252 = 50. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write down the numbers in index form. They are already products of primes.A = 24 × 32
B = 2 × 33 × 5 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: A: 24 | B: 21
Prime 3: A: 32 | B: 33
Prime 5: A: 50 | B: 51 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 1, so 2
3: smallest power is 2, so 32
5: smallest power is 0, so it is left out
HCF = 2 × 32 = 18 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 4, so 24
3: largest power is 3, so 33
5: largest power is 1, so 5
LCM = 24 × 33 × 5 = 2160
Check. The HCF divides every number: 144 ÷ 18 = 8, 270 ÷ 18 = 15. Every number divides the LCM: 2160 ÷ 144 = 15, 2160 ÷ 270 = 8. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one. Index form means writing repeated multiplication as a power.
Method. Compare the prime factorisations: for the HCF take the smallest power of each prime, for the LCM take the largest power of each prime.
- Write down the numbers in index form. They are already products of primes.A = 22 × 52 × 7
B = 23 × 5 × 11 - For each prime, write down its power in every number (use power 0 if it is missing). Comparing the powers prime by prime is the same as filling in a Venn diagram. A missing prime has power 0 because p0 = 1.Prime 2: A: 22 | B: 23
Prime 5: A: 52 | B: 51
Prime 7: A: 71 | B: 70
Prime 11: A: 110 | B: 111 - HCF: for each prime take the smallest power. The HCF can only use primes that all the numbers share, and no more copies than the number with the fewest.2: smallest power is 2, so 22
5: smallest power is 1, so 5
7: smallest power is 0, so it is left out
11: smallest power is 0, so it is left out
HCF = 22 × 5 = 20 - LCM: for each prime take the largest power. The LCM must contain enough of every prime for every number to divide into it.2: largest power is 3, so 23
5: largest power is 2, so 52
7: largest power is 1, so 7
11: largest power is 1, so 11
LCM = 23 × 52 × 7 × 11 = 15,400
Check. The HCF divides every number: 700 ÷ 20 = 35, 440 ÷ 20 = 22. Every number divides the LCM: 15,400 ÷ 700 = 22, 15,400 ÷ 440 = 35. ✓
Watch out. HCF: smallest powers, and leave out any prime that is missing from a number. LCM: largest powers, using every prime that appears.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Use HCF × LCM = x × the other number.
- Write the equation. HCF × LCM = product of the two numbers.6 × 90 = x × 18
- Multiply the left side. 540 = 18x
- Divide both sides by 18. This leaves x on its own.x = 540 ÷ 18 = 30
Check. HCF(30, 18) = 6 and LCM(30, 18) = 90. ✓
Watch out. Check your answer really has this HCF and LCM, because more than one value can pass the product test.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Use HCF × LCM = x × the other number.
- Write the equation. HCF × LCM = product of the two numbers.4 × 84 = x × 28
- Multiply the left side. 336 = 28x
- Divide both sides by 28. This leaves x on its own.x = 336 ÷ 28 = 12
Check. HCF(12, 28) = 4 and LCM(12, 28) = 84. ✓
Watch out. Check your answer really has this HCF and LCM, because more than one value can pass the product test.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Use HCF × LCM = x × the other number.
- Write the equation. HCF × LCM = product of the two numbers.5 × 150 = x × 30
- Multiply the left side. 750 = 30x
- Divide both sides by 30. This leaves x on its own.x = 750 ÷ 30 = 25
Check. HCF(25, 30) = 5 and LCM(25, 30) = 150. ✓
Watch out. Check your answer really has this HCF and LCM, because more than one value can pass the product test.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. Find the HCF and LCM from the powers, multiply them, and compare with a × b.
- Find the HCF. Smallest power of each prime: 21, 31, 50.HCF = 2 × 3 = 6
- Find the LCM. Largest power of each prime: 23, 32, 51.LCM = 23 × 32 × 5 = 360
- Multiply the HCF and LCM. Add the powers (index law).6 × 360 = 2160. In index form: 2 × 3 × 23 × 32 × 5 = 24 × 33 × 5
- Multiply a and b. Work out the two numbers first: a = 24 and b = 90.24 × 90 = 2160
- Compare. For each prime, the smaller power plus the larger power is the sum of the two powers.Both equal 2160.
Check. 24 × 33 × 5 = 16 × 27 × 5 = 2160. ✓
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The number of days must be a multiple of 6, of 8 and of 9. We want the smallest: the LCM of three numbers. Do two numbers first, then bring in the third.
- Find the LCM of 6 and 8. List multiples. 6: 6, 12, 18, 24. 8: 8, 16, 24.LCM = 24
- Find the LCM of 24 and 9. List multiples of 24: 24, 48, 72. Test each for 9: 24 ÷ 9 and 48 ÷ 9 leave remainders, 72 ÷ 9 = 8.LCM = 72
- Answer. Check that 72 works for all three.72 ÷ 6 = 12, 72 ÷ 8 = 9, 72 ÷ 9 = 8. They next meet after 72 days.
Check. Prime factors: 6 = 2 × 3, 8 = 23, 9 = 32. Largest powers: 23 × 32 = 72. ✓
Watch out. Find the LCM of two numbers, then the LCM of that answer with the third.
Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.
Method. The product of the two numbers is HCF × LCM. Find factor pairs of that product that fit the other clues.
- Use HCF × LCM = product of the numbers. 5 × 60 = 300, so the numbers multiply to 300
- List the pairs of numbers between 10 and 40 that multiply to 300. Test factor pairs of 300.10 × 30, 12 × 25, 15 × 20
- Test the HCF of each pair. We need an HCF of 5.10 and 30 have HCF 10 ✗
12 and 25 have HCF 1 ✗
15 and 20 have HCF 5 ✓ - Check the LCM. 15 = 3 × 5 and 20 = 22 × 5.LCM = 22 × 3 × 5 = 60 ✓
Check. 15 and 20 are both between 10 and 40, HCF = 5 and LCM = 60. ✓