Non zero digit

[didieravoaka]

Please help!

Thanks.

[ceilidh.erickson]

When you have a digits question like this, you can represent the digits as variables (I'm going to use X and Y, since I don't know how to copy those weird symbols here!):

The two-digit number XY could be represented as 10X + Y, and YX would be 10Y + X. For example, the number 23 is (10)(2) + (3).

Rewrite this problem as (10X + Y)^2 - (10Y + X)^2. This is a difference of squares, so we can factor:
((10X + Y) + (10Y + X))((10X + Y) - (10Y + X))
(11X + 11Y)(9X - 9Y)
(11(X + Y))(9(X - Y))
99(X + Y)(X - Y)

We're told that this number is a perfect square. In order for 9*11 times some product (X + Y)(X - Y) to be a perfect square, then (X + Y)(X - Y) must at least have a factor of 11. (Because 9 is already a square, it doesn't need a factor of 3). The only single digits that would produce a factor of 11 (but no other factors) would be (6 + 5)(6 - 5).

Thus, our product is (9)(11)(11) = 1089.

[ceilidh.erickson]

In the last step, if you wanted to shortcut the computation, we could say that the product = (33)^2.

The units digit of (33)^2 must be a 9, so the only possible answer is E.

[Matt@VeritasPrep]

Just for fun, here's an approach that avoids algebra.

(" Â¥)² - (Â¥" )² = x²

Notice that " Â¥ and Â¥"  have the same digits. That means that they must have the same digit SUM (i.e. "  + Â¥), so they must have the same remainder when divided by 9. (Remember that we add the digits of a number to check whether a number is divisible by 9.)

Since they each have the same remainder when divided by 9, (" Â¥)² - (Â¥" )² must have a remainder of 0 when divided by 9. So (" Â¥)² - (Â¥" )² is divisible by 9, which means x² is also divisible by 9.

So our answer must be divisible by 9, leaving only two options: 576 and 1089. Now remember that we use the same trick to check divisibility by 11: we add the digits, etc. So our answer must also be divisible by 11, albeit for slightly more complicated reasons. From here 576 is out and 1089 is the pick.