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Question

The arithmetic mean of two distinct positive integers x and y is a two-digit integer. The geometric mean of x and y is obtained by reversing the digits of the arithmetic mean. What is |x - y|?

Accepted Answer

Let the 2 digit number be 10a+b. We know x+y=20a+2b and xy = 10b+a. Thus. x^2+2xy+y^2 = 400a^2+80ab+4b^2 and xy = 100b^2+20ab+a^2. Notice that |x-y| = (x-y)^2. We know that (x-y)^2 = (x+y)^2-4xy = 396a^2-396b^2. Then, |x-y| = 611 b^2-a^2. Clearly, |x-y| must be a multiple of 11, so the only possible answer is 66.