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If a and b are positive integers, is a^2-b^2 divisible by 4?

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by Max@Math Revolution » Mon Jun 24, 2019 12:07 am

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[GMAT math practice question]

If a and b are positive integers, is a^2-b^2 divisible by 4?

1) a+b is divisible by 4
2) a^2+b^2 has remainder 2 when it is divided by 4
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Source: — Data Sufficiency |
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by Max@Math Revolution » Tue Jun 25, 2019 11:51 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. We should simplify the conditions, if necessary.

Condition 1)
If a + b is divisible by 4, then a^2 - b^2 = (a+b)(a-b) is divisible by 4.
Thus, condition 1) is sufficient.

Condition 2)
The squares of 1, 2, 3, 4, ... are 1, 4, 9, 16, ..., respectively and they have remainders of 1, 0, 1, 2, ... , respectively, when they are divided by 4.
Thus, if a^2 + b^2 has remainder 2 when it is divided by 4, both a and b are odd integers.
This implies that both a + b and a - b are even integers, and a^2 - b^2 = ( a + b )( a - b ) is divisible by 4.
Thus, condition 2) is sufficient too.

Therefore, D is the answer.
Answer: D

This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.
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