abc are integers if gcdab1 ab bc then prove abcSolutiongcdab

a,b,c are integers, if gcd(a,b)=1, a|b, b|c, then prove ab|c

Solution

gcd(a,b) = 1 , a|b , b|c

ab|c?

Since gcd(a,b) = 1 and according to Bezout\'s lemma, then there exist integers s and t such that:

as + bt = gcd(a,b) = 1

Multiply both sides by c to obtain:

cas + cbt = c

Since a|c, there exist an integer m such that c = ma. Since b|c, there exist an integer n such that c = nb.

Substitute these expressions for c in the previous equation:

nbas + mabt = c

Common factor:

ab(ns + mt) = c

Since ab divides the left hand side, it must also divide the right hand side.

Then ab|c.