Competition problem — Number theory

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Number theory Difficulty 5.0 AIME, harder Find the answer

Can $x$ , $\frac{14 x+5}{9}$ , and $\frac{17 x-5}{12}$ all be integers at the same time?

Solution

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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.