Prove that the product of three consecutive positive integers is divisible by 6.

Let n is any positive integer.

Since any positive integer is of the form 6p or, 6p + 1 or, 6p + 2 or, 6p + 3 or, 6p + 4 or, 6p + 5.


Case 1:


If n = 6p then


n(n + 1)(n + 2) = 6p (6p + 1)(6p + 2), which is divisible by 6.


Case 2:


If n = 6p + 1, then


n(n + 1)(n + 2) = (6p + 1)(6p + 2) (6p + 3) 6 (6p + 1)(3p + 1)(2p + 1), which is divisible by 6.


Case 3:


If n = 6p + 2, then


n(n + 1)(n + 2) = (6p + 2) (6p + 3)(6p + 4) 12 (3p + 1)(2p + 1)(2p + 3), which is divisible by 6.


Similarly, n(n + 1)(n + 2) is divisible by 6 if n = 6p + 3 or, 6p + 4 or, 6p + 5.


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