**Theorem.** A subgroup of a cyclic group is cyclic.

*Proof.* Let be a cyclic group generated by and let . If then is cyclic. If , then for . Let be the smallest integer in such that .

I claim that generates ; that is, . We must show that every is a power of . Let . Then since , for some positive integer . By the division algorithm for , there exist such that where . Then so . We must have that because is a group and . Since is the smallest positive integer such that and , it must be the case that . Thus, , , and so is a power of .

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