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

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[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.

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by fskilnik@GMATH » Thu Dec 13, 2018 5:02 am

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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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Brent@GMATPrepNow » Thu Dec 13, 2018 5:35 am

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Max@Math Revolution wrote: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.
Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN


Target question: Is mn an even number?

Given: m and n are positive integers

Statement 1: m/n is an even number.
m/n = some even integer
Multiply both sides of the equation by n to get: m = (some even integer)(n)
This means m MUST be EVEN
If m is EVEN, then mn must be EVEN
So, the answer to the target question is YES, mn IS an even number
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: m + n is an even number
There are several values of m and n that satisfy statement 2. Here are two:
Case a: m = 2 and n = 2. Notice that 2 + 2 = 4, which is even. In this case, mn = (2)(2) = 4. So, the answer to the target question is YES, mn IS an even number
Case b: m = 1 and n = 1. Notice that 1 + 1 = 2, which is even. In this case, mn = (1)(1) = 1. So, the answer to the target question is NO, mn is NOT an even number
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Sun Dec 16, 2018 5:24 pm

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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