Is 111…11 – 222…22 a perfect square, where 111…11 and 222…22 are n and m digit number

[Max@Math Revolution]

[GMAT math practice question]

Is 111…11 – 222…22 a perfect square, where 111…11 and 222…22 are n and m digit numbers, respectively?


1) n = 2m.
2) m = 5.

[Max@Math Revolution]

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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.
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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 conditions if necessary.

111…11 – 222…22 = 999…99 / 9 – 2(999…99/9) = (10^n - 1)/9 – 2(10^m - 1) / 9
= [(10^n - 1) - 2(10^m - 1)] / 9
= [(10^n – 1 + 2*10^m + 2)] / 9
= [(10^2m – 2*10^m + 1)] / 9
= (10^m – 1)^2 / 9
= [(10^m – 1) / 3]^2, if n = 2m from condition 1).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we don’t have any information about n, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A

Since both conditions together are trivial, C is not an answer. If one condition includes a ratio and the other condition just gives a number, the condition, including the ratio is most likely to be sufficient by Tip 4. This tells us that A is most likely to be the answer to this question.