Let a b c n be integers Prove that if a n and b n with gcd

Let a, b, c, n be integers. Prove that if a | n and b | n with gcd(a, b) = 1, then ab | n. If a | bc and gcd(a, b) = 1, then a | c.

Solution

(a)

a|n means : n=ga

b|n means:   n=fb

gcd(a,b)=1 means there exist integers:x,y so that

ax+by=1

Multiplying by: n gives:

nax+nby=n

Using: n=ga=fb gives

(fb)ax+(ga)by=n

ab(fx+gy)=n

Hence, ab|n

(b)

a|bc means:

bc=ar for some integer ,r

gcd(a,b)=1 so there exist integers, x,y so that:

ax+by=1

Multiplying by c gives:

acx+bcy=c

acx+ary=c   (Using bc=ar)

a(cx+ry)=c

Hence, a|c