Originally Posted by

**pberardi** I need to see why this proof works. I have most of it I think but can someone clear me up here?

Prove that for each natural number n, 8^n = 1 mod 7

By induction show 1 is true

8 = 1 (mod 7) or 8 - 1 = 7k trivial

assume n is true there for 8^n = 1 (mod 7)

Show 8^(n+1) = 1 mod 7

multiply by 8

8*8^(n) = 8 (mod 7)

8^(n+1) = 8 (mod 7)

Now this is where I get confused. My TA said that now 8^(n+1) = 1 mod 7 because of a transitivity property. Could someone help me see this please? Is there another way of doing this that perhaps I can see it better? Thank you.