Application of Lagrange's Theorem to $Z/2018Z$

Question

I want to find the order of the subgroup of $\mathbb{Z}/2018\mathbb{Z}$ generated by 8.

I know that since the order of $\mathbb{Z}/2018\mathbb{Z}$ is finite I can use Lagrange's theorem which says that the order of the sungroup generated by 8 must divide the order of $\mathbb{Z}/2018\mathbb{Z}$ which is 2018.

Now I am not sure where to go from here besides brute force checking which divisor of 2018 is the order of the subgroup generated by 8

Is there some other theorem or fact I could use to make my life easier?

This question might help. – Théophile Nov 14 '18 at 15:28 — Théophile, Nov 14 '18 at 15:28
Can you factorize $2018$? – GNUSupporter 8964民主女神地下教會 Nov 14 '18 at 15:31 — GNUSupporter 8964民主女神地下教會, Nov 14 '18 at 15:31

score 0 · Answer 1 · answered Nov 14 '18 at 15:28

0

Hint: In a cyclic group of order $n$ generated by $g$, the subgroup generated by $g^k$ has order $n/\gcd(n,k)$.

answered Nov 14 '18 at 15:28

lhf

221,500

score 0 · Accepted Answer · answered Nov 14 '18 at 15:50

0

Hint $\,\ 2018\mid 8n\iff 1009\mid 4n\iff 1009\mid n$

answered Nov 14 '18 at 15:50

Bill Dubuque

282,220

Application of Lagrange's Theorem to $Z/2018Z$

2 Answers2