If k, n, and m are positive integers, is kn even?

(1) k = m^3

k can be odd or even depending on the value of m. Also, we have no information about n. So kn can be even or odd. INSUFFICIENT. Eliminate A and D.

(2) n = (m - 1)^3

Similar to (1), n can be odd or even depending on the value of m. Also, we have no information about n. So kn can be even or odd. INSUFFICIENT. Eliminate B.

Combining (1) and (3):

kn = \((m * (m-1))^3\), which is cube of product of two consecutive integers. Product of two consecutive integers is even => cube of even is also even => kn is even. Answer is C.

Note: kn cannot be 0 since both k and n are positive integers.

Official Explanation

We want to know whether kn is even, and both k and n are integers, so it will be sufficient to know that k or n is even, since the product of the two will therefore include a factor of 2. On to the data statements, separately first.

Statement (1) tells us about k and m. If m is even, k is even. If m is odd, k is odd. But we don't know whether m is even or odd, so we can't draw any conclusions. The even/oddness of both k and n are still unconstrained. Statement (1) is insufficient.

Statement (2) is similar to Statement (1) in that it relates one of our variables of interest (this time n) to m. If m is odd, then n is even. If m is even, then n is odd. But we don't know whether m is odd or even, and we have no constraint or conclusion about k, so this statement is also insufficient.

Combining the statements, we can combine conclusions. Statement (1) told us that k had the same even/odd-ness as m. Statement (2) told us that n had the opposite even/odd-ness as m. Therefore, we know that, regardless of whether m is even or odd, k and n will have opposite even/odd-ness. That means that one of the two will be even (while the other is odd), and their product will always be even. We can answer the question definitively, so the statements give sufficient information when taken together.

The correct answer is (C).

Video Explanation

Statement 1: (insufficient)
Nothing is known about "n"

Statement 2: (insufficient)
Nothing is known about "k"

On combining statement 1 and Statement 2

No matter what value m takes either of (m-1) or m will be even.
Also, m cannot be equal to 0. As this would make n = (-1) ....{statement 2}
So by combining we can conclusively say that k*n will be even as Even * Odd = Odd* Even = Even.

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