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Find common multiples

A common multiple is a whole number that can be expressed as a multiple of each of two or more given whole numbers; it appears in the overlapping portions of their skip-counting sequences or lists of multiples. The understanding includes identifying and generating common multiples within manageable whole-number ranges and distinguishing them from common factors, without extending to generalized least-common-multiple methods or larger-number algorithms.

Detailed Explanation: Find common multiples

A common multiple is a number that appears in the multiples of both numbers.

To find common multiples:

  1. List some multiples of the first number.
  2. List some multiples of the second number.
  3. Find the numbers that appear on both lists.

Example: Find the common multiples of 33 and 44 that are less than (40)(40).

List multiples of 33:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 393,\ 6,\ 9,\ 12,\ 15,\ 18,\ 21,\ 24,\ 27,\ 30,\ 33,\ 36,\ 39

List multiples of 44:

4, 8, 12, 16, 20, 24, 28, 32, 364,\ 8,\ 12,\ 16,\ 20,\ 24,\ 28,\ 32,\ 36

Now look for numbers that are on both lists:

12, 24, 36\boxed{12,\ 24,\ 36}

So, the common multiples of 33 and 44 that are less than (40)(40) are (12)(12), (24)(24), and (36)(36).

Check one of them: (24)(24) is a multiple of both numbers because

3×8=24and4×6=24.3 \times 8=24 \qquad\text{and}\qquad 4 \times 6=24.

Remember: common multiples are numbers in the overlapping lists of multiples. Factors are different because they are numbers that divide evenly into another number.

Learn by doing: Find common multiples

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