Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If m and n are positive integers, is mn an even number?

1) m/n is an even number.
2) m + n is an even number.

$$m,n\,\, \ge \,\,1\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$
$$mn\,\,\,\mathop = \limits^? \,\,\,{\text{even}}\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\boxed{\,?\,\,\,:\,\,\,m\,\,{\text{or}}\,\,n\,\,{\text{even}}\,\,}$$
$$\left( 1 \right)\,\,{m \over n} = {\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,m = n \cdot {\rm{even}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,m\,\, = {\mathop{\rm int}} \, \cdot {\rm{even}}\,\,{\rm{ = }}\,\,{\rm{even}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,m + n\,\, = \,\,{\rm{even}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Modifying the question:
mn is an even number precisely when at least one of m and n is even. So,
the question asks if either m or n is even.

Condition 1:
If m/n = 2k for some integer k, then m = 2kn, which is an even number.
Thus, condition 1) is sufficient.

Condition 2)
If m = 2 and n = 4, then m + n = 6 is even, and mn = 8 is an even number, so the answer is 'yes'.
If m = 1 and n = 3, then m + n = 4 is even, and mn = 3 is not an even number, so the answer is 'no'.
Since it does not give us a unique answer, condition 2) is not sufficient.

Therefore, the correct answer is A.
Answer: A