Year 6: HCF and LCM using prime factors
Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page
HCF and LCM using prime factors
Prime factors give us another way to find the HCF and the LCM. Write each number as a product of prime factors in index form. Then:
HCF: take the primes that every number shares, using the smallest power of each.
LCM: take every prime that appears in any number, using the largest power of each.
The Venn diagram shows the prime factors of 72 and 84. The numbers in the middle are the prime factors they share. The shared part gives the HCF. For the LCM, multiply everything in the diagram once: the middle, then the parts outside it.
Key ideas
- HCF (prime factors): the primes that every number shares, each with the smallest power.
- LCM (prime factors): every prime that appears in any number, each with the largest power.
- The HCF is never bigger than either number. The LCM is never smaller than either number.
- In index form the shared prime is easy to spot: 22 × 3 and 23 both use 2, so the HCF uses 22, the smaller power.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.24 = 23 × 3
60 = 22 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 3, 2, so use 22
3: powers 1, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 24 and 60 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.24 = 23 × 3
60 = 22 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 3
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 23 × 3 × 5 = 120
Check. 120 ÷ 24 = 5 and 120 ÷ 60 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
30 = 2 × 3 × 5
45 = 32 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.3: powers 1, 1, 2, so use 3
- Multiply them together. This product is the HCF.HCF = 3
Check. 12 and 30 and 45 are both divisible by 3. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Watch out
Do not mix up the two rules. The HCF uses the smallest power, the LCM uses the largest power. Do not multiply every prime in the HCF: only the primes that every number has are used.Exercise 3E HCF and LCM using prime factors
Non-calculatorKey words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.20 = 22 × 5
30 = 2 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 5
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, so use 2
5: powers 1, 1, so use 5 - Multiply them together. This product is the HCF.HCF = 2 × 5 = 10
Check. 20 and 30 are both divisible by 10. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.28 = 22 × 7
42 = 2 × 3 × 7 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 7
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, so use 2
7: powers 1, 1, so use 7 - Multiply them together. This product is the HCF.HCF = 2 × 7 = 14
Check. 28 and 42 are both divisible by 14. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
48 = 24 × 3 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 4, so use 22
3: powers 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 36 and 48 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.45 = 32 × 5
75 = 3 × 52 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 3, 5
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.3: powers 2, 1, so use 3
5: powers 1, 2, so use 5 - Multiply them together. This product is the HCF.HCF = 3 × 5 = 15
Check. 45 and 75 are both divisible by 15. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.33 = 3 × 11
44 = 22 × 11 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 11
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.11: powers 1, 1, so use 11
- Multiply them together. This product is the HCF.HCF = 11
Check. 33 and 44 are both divisible by 11. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.6 = 2 × 3
8 = 23 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 3 - Multiply them together. This product is the LCM.LCM = 23 × 3 = 24
Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.10 = 2 × 5
15 = 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 2
3: largest power 3
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 2 × 3 × 5 = 30
Check. 30 ÷ 10 = 3 and 30 ÷ 15 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 22 × 32 = 36
Check. 36 ÷ 12 = 3 and 36 ÷ 18 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.14 = 2 × 7
21 = 3 × 7 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 2
3: largest power 3
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 2 × 3 × 7 = 42
Check. 42 ÷ 14 = 3 and 42 ÷ 21 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.20 = 22 × 5
35 = 5 × 7 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
5: largest power 5
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 22 × 5 × 7 = 140
Check. 140 ÷ 20 = 7 and 140 ÷ 35 = 4. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
60 = 22 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 2, so use 22
3: powers 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 36 and 60 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
60 = 22 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 32
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 22 × 32 × 5 = 180
Check. 180 ÷ 36 = 5 and 180 ÷ 60 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.84 = 22 × 3 × 7
30 = 2 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 3
5: largest power 5
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 22 × 3 × 5 × 7 = 420
Check. 420 ÷ 84 = 5 and 420 ÷ 30 = 14. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32
24 = 23 × 3 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, 3, so use 2
3: powers 1, 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 2 × 3 = 6
Check. 12 and 18 and 24 are both divisible by 6. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32
24 = 23 × 3 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 23 × 32 = 72
Check. 72 ÷ 12 = 6 and 72 ÷ 18 = 4 and 72 ÷ 24 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.18 = 2 × 32
24 = 23 × 3 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 23 × 32 = 72
Check. 72 ÷ 18 = 4 and 72 ÷ 24 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Find both answers, then multiply them and compare with the product of the two numbers.
- Find the HCF of 42 and 70. Use the common factors.HCF = 14
- Find the LCM of 42 and 70. Use the lists of multiples.LCM = 210
- Multiply the HCF and the LCM. Then multiply the two original numbers, and compare.14 × 210 = 2,940
42 × 70 = 2,940
Check. 2,940 = 2,940, so they are equal. ✓
Watch out. The two results match for any two whole numbers. Use this to check your HCF and LCM.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Use the rule: the HCF times the LCM is equal to the product of the two numbers. Then find the missing number.
- Multiply the HCF and the LCM. This gives the product of the two numbers.4 × 60 = 240
- Divide by the number you know. The missing number is the product divided by 12.240 ÷ 12 = 20
- Check the HCF and LCM. Both must match the information in the question.HCF of 12 and 20 = 4, LCM = 60
Check. 12 and 20 have HCF 4 and LCM 60. ✓
Watch out. Check the answer. A number that does not give the right HCF and LCM is not the answer.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Find the HCF of each number with 36. Keep the numbers whose HCF is 9.
- Test the multiples of 9 below 50. The HCF must be 9, so each number is a multiple of 9.9, 18, 27, 36, 45
- Find the HCF of each one with 36. Keep only the ones with HCF 9.9 and 36: HCF 9
18 and 36: HCF 18
27 and 36: HCF 9
36 and 36: HCF 36
45 and 36: HCF 9 - Write the numbers that pass. These are the answers.9, 27, 45
Check. 9 ÷ 9 = 1, 27 ÷ 9 = 3 and 45 ÷ 9 = 5. ✓
Watch out. 18 and 36 are multiples of 9, but their HCF with 36 is bigger than 9.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.Try 8 and 12: HCF 4, LCM 24, and 4 × 24 = 96 = 8 × 12. Try 9 and 15: HCF 3, LCM 45, and 3 × 45 = 135 = 9 × 15.
- Explain why. Each prime is shared out between the HCF and the LCM: the HCF uses the smaller power and the LCM uses the larger power. Together they use each original power once.Always
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Both numbers are multiples of their HCF, 6. Try the pairs of multiples of 6 that add up to 42.
- List the pairs that add up to 42. Both numbers must be multiples of 6.6 + 36 = 42
12 + 30 = 42
18 + 24 = 42 - Check the HCF and LCM of each pair. Only one pair has an LCM of 72.6 and 36: HCF 6, LCM 36
12 and 30: HCF 6, LCM 60
18 and 24: HCF 6, LCM 72 - Choose the pair that works. Its HCF is 6 and its LCM is 72.18 and 24
Check. 18 + 24 = 42, HCF = 6 and LCM = 72. ✓
Watch out. The HCF divides into both numbers, so both numbers are multiples of 6.