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 $\boxed{D: \text{Not Finite}}$

1982 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

Page

Toolbox

Search

1982 AHSME Problems/Problem 20

Contents

1982 AHSME Problems/Problem 20

Problem

Solution

See Also