If a and b are positive integers, and (2^3)(3… | GMAT Sprint

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PS · Problem Solving

705-805

Divisibility/Multiples/Factors

If a and b are positive integers, and $(2^3)(3^4)(5^7) = (a^3)*b$ , how many different possible values of b are there?

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Official Explanation

Official

First, understand that this is a very hard question. It involves several sophisticated concepts, including prime factorization and counting.

Insight #1: The question asks for number of possible values of

b

, but it's much easier to count the number of possible values for

a^3

. Every value of

a^3

will be paired with a unique value of

b

, so we will count the

a^3

's as a way to get the answer.

Insight #2:

a^3

is a perfect cube, and every prime factor in a perfect cube must appear either three times or some number of times that is a multiple of three. Thus,

a^3

could be 1, and all the factors could be in

b

.

- If

a^3

has any factors of 2, it only could have

2^3

, because that's all the factors of two available.

- If

a^3

has any factors of 3, it only could have

3^3

, because the available powers of 3 don't go up as high as any other multiple of 3.…

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What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.

If a and b are positive integers, and (23)(34)(57)=(a3)∗b(2^3)(3^4)(5^7) = (a^3)*b(23)(34)(57)=(a3)∗b, how many different possible values of b are there?