2021 Fall AMC 10A Problems/Problem 8

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?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution 1

Note that the number $\overline{xy} = 10x + y.$ By the problem statement, $10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1).$ 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 $89.$ Thus, the answer is $\boxed{\textbf{(B).}}$

~NH14

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2021 Fall AMC 10A Problems/Problem 8

Solution 1