A multiple is an integer that can divide evenly into another integer. If c /d is an integer, then c is a multiple of d. The numbers 45, 27, and 18, for example, are all multiples of 9. Here’s a better example: What are some multiples of 4? The numbers 12, 20, and 96 are all multiples of 4. How do we know? Because
Also, note that any integer, n, is a multiple of 1 and n, because 1 × n = n.
Least Common Multiple
The least common multiple (LCM) of two integers is the smallest multiple that the two numbers have in common. The LCM of two numbers is, like the GCF, useful when manipulating fractions. Also similar to the GCF, you can’t find the LCM without using prime factorization. For example, what’s the least common multiple of 4 and 6? Begin by prime factorizing:
The LCM—get this, it’s tricky—is equal to the multiplication of each factor by the maximum number of times it appears in either number. Since 2 appears twice in the prime factorization of 4, it will appear twice (2 × 2) in the LCM. Since 3 appears once, it will appear once. So the LCM of 4 and 6 is 2 × 2 × 3 = 12.
One more example will help. What is the LCM of 14 and 38? Prime factorize:
Since 2 appears a maximum of once in either number, it will appear once in the LCM. Same goes for 7 and 19, making the LCM 2 × 7 × 19 = 266.
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