Let be a homomorphism. Let be an element of of period 10. Prove that must have period 1 or 2 or 5 or 10.
I can prove that may have period 10.
because have period 10
by property of homomorphism
Therefore may have period 10.
Suppose that has period . Then by the division algorithm, we can write where are integers and . Then, following your original calculation,
First note that as was the period. This means that . But since , we must take (i.e. was the smallest nonzero number such that ). This means that , so the period must divide 10. So we can conclude that .