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Question

A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

Accepted Answer

Note that the number xy = 10x + y. By the problem statement, 10x + y = x + y^2 9x = y^2 - y 9x = y(y-1). From this we see that y(y-1) must be divisible by 9. This only happens when y=9. Then, x=8. Thus, there is only 1 cuddly number, which is 89.