Let and be two 2-digit numbers such that is obtained by reversing the digits of . Suppose they also satisfy for some positive integer . Find .
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Let x = a b , y = b a x = 1 0 a + b , y = 1 0 b + a
Subbing in, we have
9 9 a 2 − 9 9 b 2 = m 2 9 9 ∣ m ⇒ 1 0 8 9 ∣ m Let m = 1 0 8 9 k 2 , k ∈ N . 9 9 a 2 − 9 9 b 2 = 1 0 8 9 k 2 a 2 − b 2 = 1 1 k 2 ( a − b ) ( a + b ) = 1 1 k 2 1 1 ∣ ( a − b ) (rej) or 1 1 ∣ a + b Clearly, a + b < 2 0 ⇒ a + b = 1 1 ⇒ a − b = k 2
Setting k = 1 , 2 , 3 , we find that the unique solution is a = 6 , b = 5 . Hence, we have x + y + m = 6 5 + 5 6 + 3 3 = 1 5 4 .