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.