megafan wrote:

If k is a multiple of 3 and \(k = (m^2)n\), where m and n are prime numbers, which of the following must be a multiple of 9?

(A) \(m^2\)

(B) \(n^2\)

(C) \(mn\)

(D) \(mn^2\)

(E) \((mn)^2\)

Source: Gmat Hacks 1800

Given prime factorization of k:

\(k = (m^2)n\),

If k is a multiple of 3, we can say that either m = 3 or n = 3.

So either m^2 or n^2 will be a multiple of 9 but we don't know which of them is a multiple of 9.

That is why none of A, B, C and D work.

(E) \((mn)^2\) includes both\(m^2\) and \(n^2\) and hence it must be a multiple of 9.

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Karishma

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