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I was doing some practice problems for my exam next week and I could not figure this out. (This is my first post on PF btw)

2. Homework Equations : Let a be element of order n in group and let k be a positive integer. Then <a^k> = <a^gcd(n,k)> and |a^k| = n/gcd(n,k).

3. Attempt at solution:

10 = n/gcd(n, 28); 20 = n/gcd(n, 22)

Setting n equal to each other, 10gcd(n, 28) = 20gcd(n,22)

gcd(n, 28) = 2gcd(n, 22)

The possible values for n are 4, 8, 12, 16, 20, 24, ... , so on.

Not sure where to go from here.