Hello, R.Laramee!

Here one approach . . .If 6 divides for all , prove for

What I have so far is: .

. . which I have worked out to: .

I know , but how to I prove 6 divides the rest?

Actually, we know that: . is divisible by 6.

. . So we are concerned with: .

Note that: .

. . This is: .

With two consecutive integers, one must be even and the other must be odd.

Hence, iseven. . . Therefore, is a multiple of 6.