Track / Stage 5 / 12 of 400 #612 of 1964

Problem 612

AIME late

Number theory Difficulty 5.0 Find the answer

1. $[\mathbf{3}]$ If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$ , find $a+b$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Answer:
47
Solution: We can factor the equation as $\left(a-b^{2}\right)\left(a+b^{2}\right)=41 \cdot 49$ , from which it is evident that $a=45$ and $b=2$ is a possible solution. By examining the factors of 2009 , one can see that there are no other solutions.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.