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Problem 624

AIME late

Number theory Difficulty 5.1 Find the answer

Folklore

It is known that $b=2013^{2013}+2$ . Will the numbers $b^{3}+1$ and $b^{2}+2$ be coprime?

Your answer

Official solution

The number 2013 is divisible by 3, since the sum of its digits equals 6. Therefore, $b^{3} \equiv-1(\bmod 3)$ . This means, $b^{3}+1 \equiv-1+1=0$ $(\bmod 3)$ and $b^{2}+2 \equiv 1+2=0(\bmod 3)$ .

Thus, these numbers have a common divisor of 3.

## Answer

They will not.

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