Competition problem — Number theory

Number theory Difficulty 5.7 AIME, harder Prove it Russia

Is it possible to replace the stars in the equality
$\{\text{l.c.m.}\}(*, *, *) - \{\text{l.c.m.}\}(*, *, *) = 2009$
by six consecutive positive integers (but not necessarily in the successive order) so that the equality would be valid? (R. Zhenodarov)

Solution

It is not possible.

Suppose such numbers exist. The least common multiple (LCM) of several numbers is divisible by each of them and, therefore, by each of their divisors. Thus, if among the numbers for which the LCM is taken there is an even number, then the LCM will also be even. Since $2009$ is an odd number, one of the two LCMs must be odd; therefore, all even numbers must be in one LCM.

Among six consecutive natural numbers, there are three even and three odd numbers, so one LCM will be taken from three consecutive even numbers, and the other from three consecutive odd numbers. But in any such triple, there will be a number divisible by $3$ . Then both LCMs are divisible by $3$ , and their difference is divisible by $3$ . But $2009$ is not divisible by $3$ —contradiction.

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