" and ¥ represent nonzero digits, and (" ¥)² - (¥" )² is a perfect square. What is that perfect square?
121
361
576
961
1089
a² - b² = (a+b)(a-b).
Thus:
(" ¥)² - (¥" )² = (" ¥ + ¥" )(" ¥ - ¥" ).
One approach is to TEST cases and look for a PATTERN.
Case 1: " ¥=21, ¥" =12
21² - 12² = (21+12)(21-12) = 33*9 =
3*3*3*
11.
Case 2: x=52, ¥" =25
52² - 25² = (52+25)(52-25) = 77*27 =
3*3*3*7*
11.
Case 3: x=73, ¥" =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.
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