Competition problem — Number theory

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Number theory Difficulty 5.9 AIME, harder Prove it

25. Let $a, b$ be positive integers, $(a, b)=1$ ; and let $\rho$ be a real number. Prove: If $a \rho, b \rho$ are integers, then $\rho$ is also an integer.

Solution

25. Let's assume $\rho \neq 0$ . Set $a \rho=c_{1}, b \rho=c_{2}$ , then we have $a c_{2}=b c_{1}$ . Using the condition $(a, b)=1$ , we get $a\left|c_{1}, b\right| c_{2}$ .

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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.