1982 AHSME Problems/Problem 20

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1982 AHSME Problems/Problem 20

Problem

The number of pairs of positive integers $(x,y)$ which satisfy the equation $x^2+y^2=x^3$ is

$\text {(A)} 0 \qquad  \text {(B)} 1 \qquad  \text {(C)} 2 \qquad  \text {(D)} \text{not finite} \qquad  \text {(E)} \text{none of these}$

Solution

Rearrange the equation to $y^2=x^3-x^2=x^2(x-1)$. This equation is satisfied whenever $x-1$ is a perfect square. There are infinite possible values of $x$, and thus the answer is $\boxed{D: \text{Not Finite}}$

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