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(†¥)² - (¥†)² is a perfect square. What is the perfe

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(†¥)² - (¥†)² is a perfect square. What is the perfe

by funky_munky » Thu Jun 12, 2014 11:35 am
Could someone please help me solve this question. I have the explanation to the solution, however, I don't understand how to solve it.

Thanks!

Question:

"  and Â¥ represent nonzero digits, and (" Â¥)² - (Â¥" )² is a perfect square. What is that perfect square?

121

361

576

961

1089

Explanation for Solution:

Let's begin by representing the two digit number " Â¥ as 10"  + Â¥ and the two digit number Â¥"  as 10Â¥ + " . We have (10"  + Â¥)² - (10Â¥ + " )², which simplifies as 99 * (" Â² - ¥²). 99 = 3 * 3 * 11, so our answer must divide by 9 and 11, and 1089 is the only answer choice meeting such a condition.

Just for fun, if we wanted to go further and actually find "  and Â¥, we could notice that (" Â² - ¥²) must divide by 11, which means that either ("  + Â¥) or ("  - Â¥) must = 11. Since "  and Â¥ are single digits, ("  + Â¥) must = 11. ("  - Â¥) must also equal a perfect square, so "  - Â¥ must equal 4 or 1. ("  + Â¥) =11 and ("  - Â¥) = 4 is a system without integer solutions, however, so we must have ("  + Â¥) = 11 and ("  - Â¥) = 1, or "  = 6 and Â¥ = 5. Notice that 65² - 56² = (65+56)(65-56) = 121*9 = 1089. Success!
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by GMATGuruNY » Thu Jun 12, 2014 1:27 pm
"  and Â¥ represent nonzero digits, and (" Â¥)² - (Â¥" )² is a perfect square. What is that perfect square?

121

361

576

961

1089
a² - b² = (a+b)(a-b).
Thus:
(" Â¥)² - (Â¥" )² = (" Â¥ + Â¥" )(" Â¥ - Â¥" ).

One approach is to TEST cases and look for a PATTERN.

Case 1: " Â¥=21, Â¥" =12
21² - 12² = (21+12)(21-12) = 33*9 = 3*3*3*11.

Case 2: x=52, Â¥" =25
52² - 25² = (52+25)(52-25) = 77*27 = 3*3*3*7*11.

Case 3: x=73, Â¥" =37
73² - 37² = (73+37)(73-37) = 110*36 = 2*2*2*3*3*5*11.

In every case, the resulting prime-factorization includes 3*3*11.
Implication:
The correct answer choice must be divisible by both 3*3=9 and 11.

An integer is divisible by 9 only if the sum of its digits is a multiple of 9.
A: 1+2+1 = 4, which is not a multiple of 9. Eliminate A.
B: 3+6+1 = 11, which is not a multiple of 9. Eliminate B.
C: 5+7+6 = 18, which is a multiple of 9. Hold onto C.
D: 9+6+1 = 16, which is not a multiple of 9. Eliminate D.
E: 1+0+8+9 = 18, which is a multiple of 9. Hold onto E.

Check whether answer choice C is divisible by 11.
C: 576 = 9*69 = 3*3*3*23, which is not a multiple of 11. Eliminate C.

The correct answer is E.
Last edited by GMATGuruNY on Fri Jun 17, 2016 8:17 am, edited 1 time in total.
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by [email protected] » Thu Jun 12, 2014 10:13 pm
Hi funky_munky,

This question is tougher than a typical GMAT "symbolism" question (most symbolism question are based around basic arithmetic or algebra) and whoever wrote it didn't use proper phrasing (the question should ask "Which of the following COULD be that perfect square?"

The logic behind this prompt is built around some rarer arithmetic Number Property rules....

First off, the prompt can be re-written as X^2 - Y^2 = a perfect square (note that X and Y are both 2-digit numbers with none of the digits as 0 and the two numbers are "mirrors" of one another e.g. 14 and 41).

X^2 - Y^2 = (X + Y)(X - Y)

Now, as to the Number Properties:

1) If you add two "mirrored" 2-digit numbers, then you ALWAYS get a multiple of 11.

eg
14 + 41 = 55.....a multiple of 11
27 and 72 = 99....a multiple of 11
87 and 78 = 165....a multiple of 11

2) If you subtract two "mirrored" 2-digit numbers, then you ALWAYS get a multiple of 9.

41 - 14 = 27...a multiple of 9
72 - 27 = 45...a multiple of 9
87 - 78 = 9...a multiple of 9

This ultimately means that the final answer MUST be a multiple of 11 (because X + Y is a multiple of 11) AND a multiple of 9 (because X - Y is a multiple of 9).

The only answer that fits these rules is E

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