show g=5 is a generator of p=647 when p is prime

Question

How would I show that $g=5$ is a generator of $647$ using Lagrange’s theorem.

So far I have $p-1=646$, thus the $ord<g>$ = $1,2,17,19,646$,

How would I show from here using Lagrange’s theorem that $g=5$ is a generator?

Answer 1

The order of $\mathbb{Z}^\times_{647}$, as you note, is $646 = 2 \times 17 \times 19$, so any proper subgroups of $\mathbb{Z}^\times_{647}$ must have order in $S = \{2, 17, 19, 34, 38, 323\}$. Check that $5^n \not \equiv 1 \pmod {647}$ for any $n \in S$ (in fact, it suffices just to check $\{34, 38, 323\}$—why?) and you're done.