Year 6: Common multiples and the lowest common multiple
Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page
Common multiples and the lowest common multiple
A multiple of a number is in its times table. The multiples of 6 are 6, 12, 18, 24, 30… A common multiple of two numbers is in both times tables. The multiples of 10 are 10, 20, 30, 40…, so 30 is a common multiple of 6 and 10. The lowest common multiple (LCM) is the smallest one. Every common multiple is a multiple of the 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 |
Common multiples are useful when two things repeat. If one event happens every 4 minutes and another every 5 minutes, they happen together again after a time that is a common multiple of 4 and 5. The first such time is the LCM, which is 20 minutes.
To find the LCM of three numbers, first find the LCM of two of them. Then find the LCM of that answer and the third number.
Key ideas
- A multiple is an answer in a times table. A common multiple is in the times tables of two or more numbers.
- The lowest common multiple (LCM) is the smallest common multiple. Write the multiples of each number until you find the first match.
- Multiplying two numbers always gives a common multiple, but it is not always the lowest one.
- For three numbers, find the LCM of two numbers, then the LCM of that answer and the third number.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24
- Write the multiples of 12. Stop at the same number.12, 24
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 24
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 10. Count on in steps of the number. Keep going until you reach a number that is also in the other list.10, 20, 30
- Write the multiples of 15. Stop at the same number.15, 30
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 30
Check. 30 ÷ 10 = 3 and 30 ÷ 15 = 2, so 30 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 4 and 5. Use the lists of multiples of the two numbers.4: 4, 8, 12, 16, 20
5: 5, 10, 15, 20
LCM = 20 - Find the LCM of 20 and 6. Use the lists of multiples of the two numbers.20: 20, 40, 60
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
LCM = 60 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 4, 5, 6 = 60
Check. 60 ÷ 4 = 15 and 60 ÷ 5 = 12 and 60 ÷ 6 = 10. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Watch out
Do not just multiply the numbers. 6 × 8 = 48 is a common multiple of 6 and 8, but the LCM is 24, because 24 is smaller. Also, the LCM is never smaller than the bigger of the two numbers.Exercise 3B Common multiples and the LCM
Non-calculatorKey words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 5. Count on in steps of the number. Keep going until you reach a number that is also in the other list.5, 10, 15, 20, 25, 30, 35
- Write the multiples of 7. Stop at the same number.7, 14, 21, 28, 35
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 35
Check. 35 ÷ 5 = 7 and 35 ÷ 7 = 5, so 35 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 6. Count on in steps of the number. Keep going until you reach a number that is also in the other list.6, 12, 18, 24, 30
- Write the multiples of 10. Stop at the same number.10, 20, 30
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 30
Check. 30 ÷ 6 = 5 and 30 ÷ 10 = 3, so 30 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 9. Count on in steps of the number. Keep going until you reach a number that is also in the other list.9, 18, 27, 36
- Write the multiples of 12. Stop at the same number.12, 24, 36
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 36
Check. 36 ÷ 9 = 4 and 36 ÷ 12 = 3, so 36 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24, 32, 40
- Write the multiples of 20. Stop at the same number.20, 40
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 40
Check. 40 ÷ 8 = 5 and 40 ÷ 20 = 2, so 40 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 4. Count on in steps of the number. Keep going until you reach a number that is also in the other list.4, 8, 12
- Write the multiples of 6. Stop at the same number.6, 12
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 12
Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so 12 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Common multiples are in both times tables. Start with the LCM, then count on in steps of the LCM.
- Find the LCM. The first common multiple is the LCM, because it is the smallest number in both times tables.LCM of 4 and 10 = 20
- Count on in steps of 20. Each next common multiple is 20 more than the one before.20, 40, 60, 80
Check. Each one is in both times tables: 20 ÷ 4 = 5, 40 ÷ 4 = 10, and so on. ✓
Watch out. Do not count on in steps of a or b. Count on in steps of the LCM.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. A common multiple must be a multiple of both numbers. Check each option against both times tables.
- Check each number for 6. It must divide by 6 exactly.27: multiple of 6? no
36: multiple of 6? yes
40: multiple of 6? no
45: multiple of 6? no - Check each number for 9. It must also divide by 9 exactly.27: multiple of 9? yes
36: multiple of 9? yes
40: multiple of 9? no
45: multiple of 9? yes - Choose the number that passes both checks. A common multiple must pass both checks.36
Check. 36 ÷ 6 = 6 and 36 ÷ 9 = 4. ✓
Watch out. A number can be a multiple of one of the numbers without being a common multiple.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 16. Count on in steps of the number. Keep going until you reach a number that is also in the other list.16, 32, 48
- Write the multiples of 24. Stop at the same number.24, 48
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 48
Check. 48 ÷ 16 = 3 and 48 ÷ 24 = 2, so 48 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 3 and 4. Use the lists of multiples of the two numbers.3: 3, 6, 9, 12
4: 4, 8, 12
LCM = 12 - Find the LCM of 12 and 9. Use the lists of multiples of the two numbers.12: 12, 24, 36
9: 9, 18, 27, 36
LCM = 36 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 3, 4, 9 = 36
Check. 36 ÷ 3 = 12 and 36 ÷ 4 = 9 and 36 ÷ 9 = 4. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24
- Write the multiples of 12. Stop at the same number.12, 24
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 24
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 15. Count on in steps of the number. Keep going until you reach a number that is also in the other list.15, 30, 45, 60
- Write the multiples of 20. Stop at the same number.20, 40, 60
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 60
Check. 60 ÷ 15 = 4 and 60 ÷ 20 = 3, so 60 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the lowest common multiple, then count in steps of it.
- Find the LCM first. Every common multiple is a multiple of the LCM, so you only need to count in steps of the LCM.LCM of 6 and 8 = 24
- Count on in steps of 24 up to 100. Stop before you go past the limit.24, 48, 72, 96
- Count them. There are 4 common multiples up to 100.4
Check. 24 × 4 = 96, and the next one, 120, is more than 100. ✓
Watch out. Count the numbers, not the gaps. Start with the LCM itself, because it is a common multiple.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am between 60 and 90” Keep only the numbers that fit this clue.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
- Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.66, 72, 78, 84
- Use the clue: “I am a multiple of 8” Keep only the numbers that fit this clue.72
- Check the answer against every clue. It must fit all of them.“I am between 60 and 90” ✓
“I am a multiple of 6” ✓
“I am a multiple of 8” ✓
Check. 72 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 6. Count on in steps of the number. Keep going until you reach a number that is also in the other list.6, 12, 18
- Write the multiples of 9. Stop at the same number.9, 18
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 18
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2, so 18 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
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.LCM of 4 and 6 is 12, which is bigger than both. LCM of 3 and 6 is 6.
- Look for an example that breaks the statement. One example that does not fit is enough to show that the statement is not always true.The LCM of 3 and 6 is 6. That is not bigger than 6, so the statement is not always true.
- Explain why. If one number is a multiple of the other, the LCM is the bigger number itself, so it is not bigger than both numbers.Sometimes
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am more than 100” Keep only the numbers that fit this clue.101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
- Use the clue: “I am less than 115” Keep only the numbers that fit this clue.101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
- Use the clue: “I am a multiple of 4” Keep only the numbers that fit this clue.104, 108, 112
- Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.108
- Check the answer against every clue. It must fit all of them.“I am more than 100” ✓
“I am less than 115” ✓
“I am a multiple of 4” ✓
“I am a multiple of 6” ✓
Check. 108 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.