Section: Math Section 1

5) If an integer is divisible by 6 and by 9, then the integer must be divisible by which of the following?

1. 12
2. 18
3. 24
4. 36
5. 54

An integer that’s divisible by 6 has at least one 2 and one 3 in its prime factorization. An integer that’s divisible by 9 has at least two 3s in its prime factorization. Therefore, an integer that’s divisible by both 6 and 9 has at least one 2 and two 3s in its prime factorization. That means it is divisible by 2 and 3: 2 x 3 6, 3 x 3 9, and 2 x 3 x 3 = 18. It’s not necessarily divisible by 12 or 36, each of which includes two 2s in its prime factorization. You could also do this one by picking numbers. Think of a common multiple of 6 and 9 and use it to eliminate some options. 6 x 9 = 54 is an obvious common multiple—and it’s not divisible by 12 or 36, but it is divisible by 18. The least common multiple of 6 and 9 is 18, which is also divisible by 18. It looks like every common multiple of 6 and 9 is also a multiple of 18.