Factors, multiples and common values

Factors divide without a remainder

Arrange 24 counters in equal rows. The arrangements 1 × 24, 2 × 12, 3 × 8 and 4 × 6 show all the factor pairs.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Each divides 24 exactly. Five is not a factor because four groups of five use 20 and leave four.

For a one-digit factor, check the matching multiplication fact or divide and check for a zero remainder. For example, seven is a factor of 56 because 7 × 8 = 56.

Multiples are made by multiplying

The positive multiples of six begin 6, 12, 18, 24, 30, 36, …. Twenty-four is a multiple of six, and six is a factor of twenty-four. These are two ways of describing the same multiplication relationship.

The positive multiples continue; there is no largest one. The factors of a fixed positive whole number form a finite list. Do not swap the two lists.

Find what two lists share

The factors of 18 are 1, 2, 3, 6, 9, 18. Comparing them with the factors of 24 gives common factors 1, 2, 3, 6.

For common multiples of four and six, list:

Twelve and twenty-four appear in both lists. There are more if you keep going. The rows show sequences; entries in the same column do not need to match.

Check your understanding 1 Find the first shared positive multiple

What is the smallest positive number that is a multiple of both 3 and 5?

Smallest positive common multiple

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15, because 3 × 5 = 15 and 5 × 3 = 15. Smaller positive multiples of either number do not appear in both lists.

Check your understanding 2 Choose what the story asks

You have 36 beads and want equal groups with none left over. Could each group contain 4 beads? Could it contain 5? Then name two numbers of beads that could fill both bags of 4 and bags of 6 with none left over.

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Four works because 36÷4 = 9 exactly. Five leaves one, so it does not work. Twelve and twenty-four beads are two positive common multiples of four and six. The first question checks factors of 36; the second asks for common multiples.