Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Find the answer

15) How many positive integers $n$ are there for which $8 n+50$ is a multiple of $2 n+1$ ?

Pick one

Solution

15) The answer is $\mathbf{( B )}$ . Since $8 n+50=4(2 n+1)+46, 8 n+50$ is a multiple of $2 n+1$ if and only if 46 is a multiple of $2 n+1$ . The only divisor of 46 of the form $2 n+1$ with $n$ positive is 23.

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