Hello, padsinseven!

Your interpretation of that implication is correct.

Prove by induction that is even if 3 divides .

We wish to prove that: .If , then is even.

Verify . . . True!

Assume is true: . for some integer

. . and we wish to prove that: . is even.

We have: .

. . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Therefore: . is even.