division algorithm in group theory

[leekitah]

How to use division algorithm in group theory? Could any mathematicians out there guide me through either (or both preferably) of the following questions:
Use the notion of a greatest common divisor or the division algorithm in
Z to show that every subgroup of Z has the form mZ := (mk : k e Z)
for some m e Z.
Show (using the division algorithm for integers) that if G = (g) is a
cyclic group of order n, then g^t = e for some t e Z if n divides t.
Any help would be greatly appreciated.