Recall that if g epsilon Z is an integer then gZ is a commut

Recall that if g epsilon Z is an integer, then gZ is a commutative ring. Theorem 6 Suppose that n and g are nonzero integers and that g | n. Then nZ gZ is an ideal and there is a ring isomorphism gZ/nZ Z_n/g

Solution

solution-:The greatest common divisor is |n|. For suppose a positive integer a divided n. Then by defi nition
there exists an integer r such that n = ra. By hypothesis that n|m, there also exists an integer s such
that ns = m. Then

m = ns = (ra)s = (rs)a

which is the defi nition of a|m. Therefore, any divisor of n is automatically a divisor of m. Hence the
greatest positive integer which divides both m and n is the largest positive integer which divides n,
which is |n|. hence it is true