A common multiple is a number that both of two numbers divide into
Counting in fours
The multiples of 4 are the numbers you land on when you count in fours: 4, 8, 12, 16, 20, 24, 28, and on without end. Each one is 4 times a whole number: 12 = 4 × 3 and 28 = 4 × 7.
Count on to 24 in jumps of 4
Count on in jumps of 4 and every landing is ringed: 4, 8, 12, 16, 20 and 24.
Counting in sixes
The multiples of 6 come from counting in sixes: 6, 12, 18, 24, 30, and on. Each one is 6 times a whole number: 18 = 6 × 3.
Count on to 24 in jumps of 6
In jumps of 6 the landings are 6, 12, 18 and 24. Two of them, 12 and 24, were landings in the count of fours as well.
Where both counts land
Compare the two counts. 12 is in both, and so is 24: 12 = 4 × 3 = 6 × 2, and 24 = 4 × 6 = 6 × 4. A number that is a multiple of both of two numbers is a common multiple of them. Up to 30, the common multiples of 4 and 6 are 12 and 24.
The counts go on meeting after 30: at 36, 48, 60, and on, every 12. The common multiples of 4 and 6 are the multiples of 12.
Up to 30, only 12 and 24 are on the count of fours and on the count of sixes.
The first meeting
The smallest common multiple is the first number both counts land on. For 4 and 6 it is 12. It is called the Lowest Common Multiple, or LCM for short. The LCM of 4 and 6 is 12.
Multiplying the two numbers always gives a common multiple, because 4 × 6 is 4 counted six times and also 6 counted four times. It need not be the first one: 4 × 6 = 24, and the counts have already met at 12.
Three hops of 4 reach 12, and 12 is also two sixes. It comes before 24, so it is the lowest common multiple.
On both counts, not one
A common multiple must be on both counts, so test each candidate against both numbers. 16 is a multiple of 4, but 16 ÷ 6 = 2 remainder 4, so it is not a multiple of 6. 18 is a multiple of 6, but 18 ÷ 4 = 4 remainder 2.
Multiples and factors go in opposite directions. The multiples of 6 start at 6 and grow: 6, 12, 18. The factors of 6, which are 1, 2, 3 and 6, are never more than 6.
Stepping on by the LCM
The counts of 5 and 7 first meet at 35, so the common multiples of 5 and 7 are 35, 70, 105, and on, each 35 more than the one before.
Adding a common multiple does not change what is left over. 18 in groups of 5 is 3 groups with 3 left over, and in groups of 7 it is 2 groups with 4 left over. 35 is a whole number of fives and a whole number of sevens, so 18 + 35 = 53 has the same 3 left over in groups of 5 and the same 4 left over in groups of 7.
HCF and LCM · Common Multiples: Roses That Leave 3 Over in Fives and 4 Over in Sevens
QuestionA florist has fewer than 100 roses. When she puts them into bunches of 5, she has 3 roses left over. When she puts them into bunches of 7, she has 4 roses left over. (a) What is the smallest number of roses she could have? (b) What are all the possible numbers of roses she could have?
1.List the numbers that leave 4 over in bunches of 7. Start at 4 and count on in 7s: 4, 11, 18, 25, 32, and so on.
Leaving 4 over in bunches of 7: start at 4 and count on in 7s. 4, 11, 18, 25, 32, and so on. 2.List the numbers that leave 3 over in bunches of 5. Start at 3 and count on in 5s: 3, 8, 13, 18, 23, and so on.
Leaving 3 over in bunches of 5: start at 3 and count on in 5s. 3, 8, 13, 18, 23, and so on. 3.(a) The first number on both lists is 18, so the smallest number of roses is 18.
(a) The first number on both lists is 18. 4.35 is the LCM of 5 and 7: it makes 7 bunches of 5 and 5 bunches of 7. So adding 35 roses adds whole bunches of both sizes, and the roses left over stay at 3 and 4.
35 roses make 7 bunches of 5 and 5 bunches of 7, so adding 35 keeps both remainders: 18 + 35 = 53. 5.Step on from 18 by 35: 18 + 35 = 53 and 53 + 35 = 88. The next one, 88 + 35 = 123, is more than 100.
53 + 35 = 88. The next, 88 + 35 = 123, is more than 100. 6.(b) The possible numbers of roses are 18, 53 and 88. Check: 88 = 5 × 17 + 3 = 7 × 12 + 4.
(b) The possible numbers of roses are 18, 53 and 88.
Answer: (a) 18 roses; (b) 18, 53 or 88 roses
Watch out for
- Giving only 18. The question asks for all the possible numbers, and adding 35 roses, which is 7 bunches of 5 and 5 bunches of 7, keeps both remainders the same, so 53 and 88 fit as well.
- Stepping on by 5 + 7 = 12 and getting 30. 30 = 5 × 6 leaves nothing over in bunches of 5. Only a common multiple of 5 and 7 adds whole bunches of both sizes, and the smallest is 35.
Your turn
1.Which do both 4 and 6 divide into?