Wednesday, 4 March 2015

elementary number theory - d1,d2midniff[d1,d2]midn, [LCM Universal Property]

Is d1n,d2n[d1,d2]n
true ? And if yes, how can I prove it ? I recall that [d1,d2] is the least common multiple.



My tries



For the implication : Let d1n and d2n. Then, d1d2n2. By the way, since d1,d2n, we have that (d1,d2)n where (d1,d2)=:gcd and thus [d_1,d_2](d_1,d_2)\mid n^2.




Question: How can I get [d_1,d_2]\mid n from this?



For the converse, since d_1,d_2\mid [d_1,d_2]\mid n, the claim follow.

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