Hello, hoger!

Here's the first one . . .

is a multiple of 9.1) Show that is a multiple of 9, for

a) by induction

Verify , which is a multiple of 9.

Assume , for some integer

Add to both sides:

. .

. .

We must show that the right side is a multiple of 9.

The right side is: .

. .

The inductive proof is complete.

b) by direct proof.

. .

We must show that either or is a multiple of 3.

There are three cases for

. .

. . .Then: . , a multiple of 3.

. . .

. . .

. .