-
didieravoaka
- Master | Next Rank: 500 Posts
- Posts: 163
- Joined: Tue Jan 13, 2015 11:44 am
- Thanked: 2 times
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.
