Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Prove it Ukraine

Given positive integers $a$ , $b$ , $c$ , $d$ such that $a < b < c < d$ . Is it possible for the least common multiple of $a$ and $b$ to be greater than the least common multiple of $c$ and $d$ ?

Solution

Let $a = 8$ , $b = 9$ , then $[a; b] = 72$ . We will choose the other two numbers such that $[c; d] = 50$ : $c = 10$ , $d = 25$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.