MathRevolution
[GMAT math practice question]
(number properties) 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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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.
The squares of 1, 2, 3, 4, … are 1, 4, 9, 16, …, respectively and they have remainders of 1, 0, 1, 2,
if squares are divided by 4 series comes 1,0,1,0, Can you please explain how 2 came ?