Competition problem — Number theory

Library / /161 of 353

Number theory Difficulty 4.9 AIME Find the answer

An integer $n$ is chosen uniformly at random from the set $\{1,2,3, \ldots, 2023!\}$ . Compute the probability that $\operatorname{gcd}\left(n^{n}+50, n+1\right)=1$

Your answer

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

Solution

If $n$ is even, we need $\operatorname{gcd}(n+1,51)=1$ . If $n$ is odd, we need $\operatorname{gcd}(n+1,49)=1$ . Thus, the answer is $\frac{1}{2}\left(\frac{\varphi(49)}{49}+\frac{\varphi(51)}{51}\right)=\frac{265}{357}$

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