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"Suppose 2 | n and 3 | n. Then we can choose integers j and k such that n=2j and n=3k.

Therefore 6(j−k)=6j−6k=3(2j)−2(3k)= 3n − 2n = n, so 6 | n."

My "proof" is as follows:

"Suppose 2 | n and 3 | n. Then there is an integer k such that (2)(3)k=n. Thus, (6)k=n and therefore 6 | n."

If my proof is wrong, then could you tell me why? I was wondering if it's because my proof has something to do with the fact that every number can be written as a product of prime numbers and maybe I need to state this or something. Thanks in advance.