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(proof by contradiction)

Suppose there exists a product of 3 consecutive positive integers that is not divisible by 6 :
6\nmid n(n+1)(n+2)
\implies~6\nmid [n(n+1)(n+2) - 3n(n+1)]
\implies~6\nmid [n(n+1)(n+2-3)]
\implies~6\nmid [n(n+1)(n-1)]
\implies~6\nmid [(n-1)n(n+1)]

Repeating the whole argument we see that
6\nmid [(n-2)(n-1)n]
6\nmid[1\cdot 2\cdot 3]

This is a contradiction because 1\cdot 2\cdot 3=6 and 6\mid 6.The only resolution to this contradiction is that there doesn't exist a product of 3 consecutive positive integers not divisible by 6. That is, the product of any 3 consecutive positive integers is divisible by 6.