Year 5: Multiples and common multiples
Part of the Year 5 prime numbers, factors, squares and cubes chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Multiples and common multiples
A multiple of a number is an answer in that number’s times table. You find the multiples by counting on in steps of that number. Count in 4s: 4, 8, 12, 16, 20, 24… These are the multiples of 4. Each one is 4 × 1, 4 × 2, 4 × 3, and so on.
Multiples never stop. There is no biggest multiple, so we always say “the first six multiples” or “the multiples up to 50”. A number is a multiple of itself (4 = 4 × 1). In this chapter we start at the number itself, not at 0. Some times tables have patterns: every multiple of 2 is even, every multiple of 5 ends in 0 or 5, and every multiple of 10 ends in 0.
Common multiples. Look at the hundred square. The multiples of 3 are blue. The multiples of 4 are orange. Some numbers are in both times tables. They are green. They are the common multiples of 3 and 4: 12, 24, 36, 48… The smallest of them is 12. It is called the lowest common multiple, or LCM.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Finding the LCM with lists. Write the multiples of each number. Keep going until a number is in both lists. That is the LCM. For 6 and 8: 6, 12, 18, 24 and 8, 16, 24. The LCM is 24. (Try not to just multiply the two numbers: 6 × 8 = 48 is a common multiple, but not the lowest one.)
Multiples in real life. Buses, lights and bells all repeat. If one bus leaves every 6 minutes and another every 8 minutes, they leave together again when the time is a common multiple of 6 and 8. The picture shows both buses. The first time they leave together is the LCM.
Key ideas
- A multiple of a number is in its times table: you get it by multiplying the number by 1, 2, 3, 4…
- To list multiples, count on in steps of the number. Every multiple is the same distance from the one before.
- To test a number, look at the multiples just below and just above it. Or divide: if there is no remainder, it is a multiple.
- A common multiple of two numbers is in both lists. The smallest one is the lowest common multiple (LCM).
- For real-life problems with repeats (buses, lights), the answer is a common multiple.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 7 are the answers in the 7 times table. Multiply 7 by 1, 2, 3, 4 and so on.
- Multiply 7 by 1, 2, 3, … Each answer is a multiple of 7.7 × 1 = 7 7 × 2 = 14 7 × 3 = 21
7 × 4 = 28 7 × 5 = 35 7 × 6 = 42
7 × 7 = 49 7 × 8 = 56 - Write the multiples in order. The first multiple of a number is the number itself.7, 14, 21, 28, 35, 42, 49, 56
- Look at the steps. The multiples go up by 7 each time, so each one is 7 more than the one before.7, 7 + 7 = 14, …, 56
Check. The last multiple is 7 × 8 = 56, and 56 ÷ 7 = 8. ✓
Watch out. Start at 7, not at 0. And do not add a different number each time: every step is +7.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 8 is a number in the 8 times table. Find the multiples of 8 either side of the number and see if it is one of them.
- Look at the 8 times table near 56. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.8 × 7 = 56 and 8 × 8 = 64
- Compare with the number. Is it one of them?56 is equal to 56.
- Answer the question. 56 is a multiple of 8: Yes
Check. Divide: 56 ÷ 8 = 7. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 64 for 56) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 75. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 12 = 72 and 6 × 13 = 78
- Compare with the number. Is it one of them?75 is between 72 and 78.
- Answer the question. 75 is not a multiple of 6: No
Check. Divide: 75 ÷ 6 = 12 remainder 3. There is a remainder, so it is not exact. ✓
Watch out. A multiple is always in the times table. Being close (like 78 for 75) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. List the multiples of 9, and keep only the ones between 50 and 100.
- Count on in 9s until you pass 100. We need every multiple, so do not stop too soon.9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108
- Cross out the ones that are not between 50 and 100. “Between 50 and 100” does not include 50 or 100 themselves.
45, 54, 63, 72, 81, 90, 99,108 - Write the numbers that are left. 54, 63, 72, 81, 90, 99
Check. The first one, 54, is more than 50, and 45 is not. The last, 99, is less than 100, and 108 is not. ✓
Watch out. Check the first and the last numbers against the limits.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 3. Count on in 3s up to 36.3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
- List the multiples of 4. Count on in 4s up to the same number.4, 8, 12, 16, 20, 24, 28, 32, 36
- Find the numbers in both lists. These are the common multiples. They are in bold.12, 24, 36
- Write the answer. Only list the common multiples up to 36.12, 24, 36
Check. Each one is in both times tables: 12 = 3 × 4 = 4 × 3, 24 = 3 × 8 = 4 × 6. ✓
Watch out. A number must be in both lists to be a common multiple.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 6. Count on in 6s. Keep going until a number appears in both lists.6, 12, 18, 24
- List the multiples of 8. Count on in 8s up to the same number.8, 16, 24
- Find the numbers in both lists. These are the common multiples. They are in bold.24
- Pick the smallest. The lowest common multiple is the first number that is in both lists.The LCM of 6 and 8 is 24.
Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3. Both divide exactly. ✓
Watch out. The lowest common multiple is not always 6 × 8 = 48.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The beach bus leaves every 12 minutes and the zoo bus every 9 minutes.Every 12: 12, 24, 36
Every 9: 9, 18, 27, 36 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 36
- Pick the first one. The first time they are together again is the lowest common multiple.LCM of 12 and 9 = 36
- Add it to the start time. 36 minutes after 9:00 am.9:00 am + 36 minutes = 9:36 am
Check. 36 ÷ 12 = 3 and 36 ÷ 9 = 4, so both happen exactly at 36. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The red light flashs every 4 seconds and the blue light every 6 seconds.Every 4: 4, 8, 12
Every 6: 6, 12 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 12
- Pick the first one. The first time they are together again is the lowest common multiple.LCM of 4 and 6 = 12
- Write the answer. 12 seconds after the start
Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so both happen exactly at 12. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Watch out
Do not mix up multiples and factors. Multiples are bigger (or equal): the multiples of 6 are 6, 12, 18… The factors of 6 are smaller (or equal): 1, 2, 3, 6. Also, the LCM is the lowest number in both lists. Multiplying the two numbers always gives a common multiple, but not always the lowest one.Exercise 3A Multiples and common multiples
Non-calculatorKey words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 3 are the answers in the 3 times table. Multiply 3 by 1, 2, 3, 4 and so on.
- Multiply 3 by 1, 2, 3, … Each answer is a multiple of 3.3 × 1 = 3 3 × 2 = 6 3 × 3 = 9
3 × 4 = 12 3 × 5 = 15 3 × 6 = 18 - Write the multiples in order. The first multiple of a number is the number itself.3, 6, 9, 12, 15, 18
- Look at the steps. The multiples go up by 3 each time, so each one is 3 more than the one before.3, 3 + 3 = 6, …, 18
Check. The last multiple is 3 × 6 = 18, and 18 ÷ 3 = 6. ✓
Watch out. Start at 3, not at 0. And do not add a different number each time: every step is +3.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 4 are the answers in the 4 times table. Multiply 4 by 1, 2, 3, 4 and so on.
- Multiply 4 by 1, 2, 3, … Each answer is a multiple of 4.4 × 1 = 4 4 × 2 = 8 4 × 3 = 12
4 × 4 = 16 4 × 5 = 20 4 × 6 = 24 - Write the multiples in order. The first multiple of a number is the number itself.4, 8, 12, 16, 20, 24
- Look at the steps. The multiples go up by 4 each time, so each one is 4 more than the one before.4, 4 + 4 = 8, …, 24
Check. The last multiple is 4 × 6 = 24, and 24 ÷ 4 = 6. ✓
Watch out. Start at 4, not at 0. And do not add a different number each time: every step is +4.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 6 are the answers in the 6 times table. Multiply 6 by 1, 2, 3, 4 and so on.
- Multiply 6 by 1, 2, 3, … Each answer is a multiple of 6.6 × 1 = 6 6 × 2 = 12 6 × 3 = 18
6 × 4 = 24 6 × 5 = 30 6 × 6 = 36 - Write the multiples in order. The first multiple of a number is the number itself.6, 12, 18, 24, 30, 36
- Look at the steps. The multiples go up by 6 each time, so each one is 6 more than the one before.6, 6 + 6 = 12, …, 36
Check. The last multiple is 6 × 6 = 36, and 36 ÷ 6 = 6. ✓
Watch out. Start at 6, not at 0. And do not add a different number each time: every step is +6.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 8 are the answers in the 8 times table. Multiply 8 by 1, 2, 3, 4 and so on.
- Multiply 8 by 1, 2, 3, … Each answer is a multiple of 8.8 × 1 = 8 8 × 2 = 16 8 × 3 = 24
8 × 4 = 32 8 × 5 = 40 8 × 6 = 48 - Write the multiples in order. The first multiple of a number is the number itself.8, 16, 24, 32, 40, 48
- Look at the steps. The multiples go up by 8 each time, so each one is 8 more than the one before.8, 8 + 8 = 16, …, 48
Check. The last multiple is 8 × 6 = 48, and 48 ÷ 8 = 6. ✓
Watch out. Start at 8, not at 0. And do not add a different number each time: every step is +8.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. The multiples of 11 are the answers in the 11 times table. Multiply 11 by 1, 2, 3, 4 and so on.
- Multiply 11 by 1, 2, 3, … Each answer is a multiple of 11.11 × 1 = 11 11 × 2 = 22 11 × 3 = 33
11 × 4 = 44 11 × 5 = 55 11 × 6 = 66 - Write the multiples in order. The first multiple of a number is the number itself.11, 22, 33, 44, 55, 66
- Look at the steps. The multiples go up by 11 each time, so each one is 11 more than the one before.11, 11 + 11 = 22, …, 66
Check. The last multiple is 11 × 6 = 66, and 66 ÷ 11 = 6. ✓
Watch out. Start at 11, not at 0. And do not add a different number each time: every step is +11.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. Each hop on the line is the same size. Add 7 to move right, or take 7 away to move left.
- Find the size of each hop. Look at the numbers that are not hidden.0 to 7 is one hop, so each hop is 7.
- Work out each missing number. Add one hop to the number before it.7 + 7 = 14
21 + 7 = 28
35 + 7 = 42 - Check the last label. The label after 7 hops is 7 × 7.7 × 7 = 49
Check. Count in 7s: 7, 14, 21, 28, 35, 42, 49. ✓
Watch out. The labels go up in 7s, not in 1s.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 18. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 3 = 18 and 6 × 4 = 24
- Compare with the number. Is it one of them?18 is equal to 18.
- Answer the question. 18 is a multiple of 6: Yes
Check. Divide: 18 ÷ 6 = 3. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 24 for 18) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 40. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 6 = 36 and 6 × 7 = 42
- Compare with the number. Is it one of them?40 is between 36 and 42.
- Answer the question. 40 is not a multiple of 6: No
Check. Divide: 40 ÷ 6 = 6 remainder 4. There is a remainder, so it is not exact. ✓
Watch out. A multiple is always in the times table. Being close (like 42 for 40) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 54. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 9 = 54 and 6 × 10 = 60
- Compare with the number. Is it one of them?54 is equal to 54.
- Answer the question. 54 is a multiple of 6: Yes
Check. Divide: 54 ÷ 6 = 9. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 60 for 54) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 72. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 12 = 72 and 6 × 13 = 78
- Compare with the number. Is it one of them?72 is equal to 72.
- Answer the question. 72 is a multiple of 6: Yes
Check. Divide: 72 ÷ 6 = 12. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 78 for 72) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.
- Look at the 6 times table near 100. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.6 × 16 = 96 and 6 × 17 = 102
- Compare with the number. Is it one of them?100 is between 96 and 102.
- Answer the question. 100 is not a multiple of 6: No
Check. Divide: 100 ÷ 6 = 16 remainder 4. There is a remainder, so it is not exact. ✓
Watch out. A multiple is always in the times table. Being close (like 102 for 100) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. List the multiples of 8, and keep only the ones between 30 and 80.
- Count on in 8s until you pass 80. We need every multiple, so do not stop too soon.8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88
- Cross out the ones that are not between 30 and 80. “Between 30 and 80” does not include 30 or 80 themselves.
24, 32, 40, 48, 56, 64, 72,80,88 - Write the numbers that are left. 32, 40, 48, 56, 64, 72
Check. The first one, 32, is more than 30, and 24 is not. The last, 72, is less than 80, and 80 is not. ✓
Watch out. Check the first and the last numbers against the limits.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. List the multiples of 12, and keep only the ones between 50 and 100.
- Count on in 12s until you pass 100. We need every multiple, so do not stop too soon.12, 24, 36, 48, 60, 72, 84, 96, 108
- Cross out the ones that are not between 50 and 100. “Between 50 and 100” does not include 50 or 100 themselves.
48, 60, 72, 84, 96,108 - Write the numbers that are left. 60, 72, 84, 96
Check. The first one, 60, is more than 50, and 48 is not. The last, 96, is less than 100, and 108 is not. ✓
Watch out. Check the first and the last numbers against the limits.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. List the multiples of 15, and keep only the ones between 20 and 100.
- Count on in 15s until you pass 100. We need every multiple, so do not stop too soon.15, 30, 45, 60, 75, 90, 105
- Cross out the ones that are not between 20 and 100. “Between 20 and 100” does not include 20 or 100 themselves.
15, 30, 45, 60, 75, 90,105 - Write the numbers that are left. 30, 45, 60, 75, 90
Check. The first one, 30, is more than 20, and 15 is not. The last, 90, is less than 100, and 105 is not. ✓
Watch out. Check the first and the last numbers against the limits.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. Count on in 9s from 63. Add 9.
- Find 63 on the list. The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, …63 = 9 × 7
- Move to the next multiple. The next multiple is one more 9.63 + 9 = 72
- Check on the grid. On the grid, 72 is the next blue number after 63.72 = 9 × 8
Check. 72 ÷ 9 = 8. ✓
Watch out. Add 9, not 1.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. Write the multiples of 9 up to 100, then count them.
- Count on in 9s. Stop before you go past 100.9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
- Count the numbers. Count each multiple once.11 numbers
- Check the last one. The next multiple, 108, is more than 100.9 × 11 = 99
Check. 100 ÷ 9 = 11 remainder 1, so there are 11 whole times 9 goes in. ✓
Watch out. Count the multiples, not the gaps.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 9 is a number in the 9 times table. Find the multiples of 9 either side of the number and see if it is one of them.
- Look at the 9 times table near 108. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.9 × 12 = 108 and 9 × 13 = 117
- Compare with the number. Is it one of them?108 is equal to 108.
- Answer the question. 108 is a multiple of 9: Yes
Check. Divide: 108 ÷ 9 = 12. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 117 for 108) is not enough.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 3. Count on in 3s. Keep going until a number appears in both lists.3, 6, 9, 12, 15
- List the multiples of 5. Count on in 5s up to the same number.5, 10, 15
- Find the numbers in both lists. These are the common multiples. They are in bold.15
- Pick the smallest. The lowest common multiple is the first number that is in both lists.The LCM of 3 and 5 is 15.
Check. 15 ÷ 3 = 5 and 15 ÷ 5 = 3. Both divide exactly. ✓
Watch out. Make sure the number is in both lists, not just one.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 4. Count on in 4s. Keep going until a number appears in both lists.4, 8, 12
- List the multiples of 6. Count on in 6s up to the same number.6, 12
- Find the numbers in both lists. These are the common multiples. They are in bold.12
- Pick the smallest. The lowest common multiple is the first number that is in both lists.The LCM of 4 and 6 is 12.
Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2. Both divide exactly. ✓
Watch out. The lowest common multiple is not always 4 × 6 = 24.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 6. Count on in 6s. Keep going until a number appears in both lists.6, 12, 18
- List the multiples of 9. Count on in 9s up to the same number.9, 18
- Find the numbers in both lists. These are the common multiples. They are in bold.18
- Pick the smallest. The lowest common multiple is the first number that is in both lists.The LCM of 6 and 9 is 18.
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2. Both divide exactly. ✓
Watch out. The lowest common multiple is not always 6 × 9 = 54.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.
- List the multiples of 8. Count on in 8s. Keep going until a number appears in both lists.8, 16, 24
- List the multiples of 12. Count on in 12s up to the same number.12, 24
- Find the numbers in both lists. These are the common multiples. They are in bold.24
- Pick the smallest. The lowest common multiple is the first number that is in both lists.The LCM of 8 and 12 is 24.
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2. Both divide exactly. ✓
Watch out. The lowest common multiple is not always 8 × 12 = 96.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Look at the numbers in the middle, where the two circles overlap.
- Read the labels. The left circle holds multiples of 4. The right circle holds multiples of 6.Left: multiples of 4. Right: multiples of 6.
- Say what the middle means. A number in the middle is in both circles, so it is in both times tables.12, 24 and 36 are in the 4 times table and the 6 times table.
- Write the answer. They are common multiples of 4 and 6 (multiples of both).
Check. 12 = 4 × 3 = 6 × 2. ✓
Watch out. The middle of a Venn diagram is for things that fit both labels.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. The LCM is the smallest number in the middle of the diagram.
- Look at the middle. These are the common multiples.12, 24, 36
- Pick the smallest. The lowest common multiple is the smallest one.The smallest is 12.
- Check with times tables. 4 × 3 = 12 and 6 × 2 = 12
Check. No smaller number is in both lists: 4, 8 and 6 are not common. ✓
Watch out. Lowest means smallest, not biggest.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. The common multiples go up in steps of the LCM, which is 12.
- Find the step. The common multiples are 12, 24, 36, so each is 12 more than the one before.12, 24, 36
- Add 12 to the last one. 36 + 12 = 48
- Check both tables. 48 ÷ 4 = 12 and 48 ÷ 6 = 8
Check. 48 is in both lists: 4 × 12 = 48 and 6 × 8 = 48. ✓
Watch out. The next common multiple is not 36 + 4 or 36 + 6.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The hull bus leaves every 15 minutes and the Leeds bus every 20 minutes.Every 15: 15, 30, 45, 60
Every 20: 20, 40, 60 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 60
- Pick the first one. The first time they are together again is the lowest common multiple.LCM of 15 and 20 = 60
- Add it to the start time. 60 minutes after 8:00 am.8:00 am + 60 minutes = 9:00 am
Check. 60 ÷ 15 = 4 and 60 ÷ 20 = 3, so both happen exactly at 60. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The red light flashs every 4 seconds and the blue light every 5 seconds.Every 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
Every 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 20, 40, 60
- Write the answer. Only use times up to 60 seconds.20, 40 and 60 seconds
Check. 20 ÷ 4 = 5 and 20 ÷ 5 = 4, so both happen exactly at 20. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 8 is a number in the 8 times table. Find the multiples of 8 either side of the number and see if it is one of them.
- Look at the 8 times table near 120. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.8 × 15 = 120 and 8 × 16 = 128
- Compare with the number. Is it one of them?120 is equal to 120.
- Answer the question. 120 is a multiple of 8: Yes
Check. Divide: 120 ÷ 8 = 15. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 128 for 120) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 8 is a number in the 8 times table. Find the multiples of 8 either side of the number and see if it is one of them.
- Look at the 8 times table near 150. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.8 × 18 = 144 and 8 × 19 = 152
- Compare with the number. Is it one of them?150 is between 144 and 152.
- Answer the question. 150 is not a multiple of 8: No
Check. Divide: 150 ÷ 8 = 18 remainder 6. There is a remainder, so it is not exact. ✓
Watch out. A multiple is always in the times table. Being close (like 152 for 150) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. A multiple of 8 is a number in the 8 times table. Find the multiples of 8 either side of the number and see if it is one of them.
- Look at the 8 times table near 200. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.8 × 25 = 200 and 8 × 26 = 208
- Compare with the number. Is it one of them?200 is equal to 200.
- Answer the question. 200 is a multiple of 8: Yes
Check. Divide: 200 ÷ 8 = 25. There is no remainder. ✓
Watch out. A multiple is always in the times table. Being close (like 208 for 200) is not enough.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.7 × 1 = 7 7 × 2 = 14 7 × 3 = 21
7 × 4 = 28 7 × 5 = 35 7 × 6 = 42 - Compare with the answer in the question. Zainab wrote: 7, 14, 21, 27, 35, 42
- Find the mistake. The fourth number should be 7 × 4 = 28. Zainab wrote 27, one less.
- Write the correct answer. The wrong number is 27. It should be 28.
Check. Count on in 7s: 7, 14, 21, 28, 35, 42. Each step is +7. ✓
Watch out. Every step must be the same size. 21 to 27 is +6, so something is wrong.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.Multiples of 4: 4, 8, 12, 16, 20, 24
All of these are even, so they are in the 2 times table. - Explain why. A reason shows what is happening for every number, not just the ones we tried.4 = 2 × 2. So 4 × any number = 2 × (2 × that number), which is a multiple of 2. For example 4 × 7 = 28 = 2 × 14.
- Decide. Always
Check. Every multiple of 4 is also in the 2 times table, so they are all even. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.Multiples of 3: 3, 6, 9, 12, 15, 18
3, 9 and 15 are odd. 6, 12 and 18 are even. - Explain why. A reason shows what is happening for every number, not just the ones we tried.The multiples of 3 go odd, even, odd, even… because each step adds 3 (an odd number). So some are odd and some are even.
- Decide. Sometimes
Check. 3 is odd and 6 is even, so we have one of each. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a multiple of 8”. List the numbers between 80 and 100 that fit this clue.88, 96
- Use the next clue: “a multiple of 6”. Cross out every number that does not fit.
88, 96 - Check the answer against every clue. It must fit all of them.“a multiple of 8” ✓
“a multiple of 6” ✓
Check. Only 96 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Find the LCM of 6 and 9 by listing multiples. Then compare it with 6 × 9.
- List the multiples of 6. Keep going until a number is in both lists.6, 12, 18
- List the multiples of 9. 9, 18
- Find the LCM. The first number in both lists.LCM of 6 and 9 = 18
- Compare with multiplying. 6 × 9 = 54. 54 is a common multiple, but it is not the lowest.18 is less than 54, so Mia is wrong.
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2. ✓
Watch out. Multiplying the two numbers gives a common multiple, but not always the lowest one. It only works when the numbers have no common factor.