Hello, Grillakis!

Your work is excellent!

Prove that 6 divides for a nonnegative integer

What I have is this:

Basis Step:

If , then

. .

So the basis step is true since its divisible by 6.

Inductive Hypothesis:

We assume true: if , then: is divisible by 6.

We must prove that if ,

. . then: . is divisible by 6.

Inductive Step:

. . . . . .

. . . . . .

. . . . . .

Where I am stuck: is divisible by 6? . . . . Yes!

We know that is divisible by 6: .

Note that is the product oftwo consecutive integers.

Hence, one of them is even and the other is odd.

And their product is even, say,

So we have: .