If a and b are integers, is a-b an even number?

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[Math Revolution GMAT math practice question]

If a and b are integers, is a-b an even number?

1) a^2b^2 is an even number
2) a^2+2b^2 is an even number

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by fskilnik@GMATH » Thu Dec 06, 2018 4:56 am

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

If a and b are integers, is a-b an even number?

1) a^2b^2 is an even number
2) a^2+2b^2 is an even number
$$a,b\,\,{\rm{ints}}$$
$$a - b\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\mathop { \Leftrightarrow \,}\limits^{\left( * \right)} \,\,\,\,\,\boxed{\,?\,\,\,:\,\,\,\left( {a,b} \right) = \left( {{\text{odd}}\,{\text{,}}\,{\text{odd}}} \right)\,\,\,{\text{or}}\,\,\,\,\left( {a,b} \right) = \left( {{\text{even}}\,{\text{,}}\,{\text{even}}} \right)\,\,}$$

$$\left( 1 \right)\,\,{\left( {ab} \right)^2}\,\,{\rm{even}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,{a^2} + 2{b^2}\,\,{\rm{even}}\,\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,$$

$$ \Rightarrow \,\,\,\left( {\rm{E}} \right)$$


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 Max@Math Revolution » Sun Dec 09, 2018 5:47 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:
For a - b to be an even number, either both a and b must be even numbers or both a and b must be odd numbers.

Since we have 2 variables (a and b) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
From condition 2), a is an even number.
From condition 1), b might either be even or odd.

Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, the correct answer is E.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.