Prove that if a divides b and b divides a then ab and gcd a

Prove that if a divides b and b divides a, then a=b, and gcd (a, b) >0.

Solution

First note that neither a nor b can be 0, because 0 doesn\'t divide any numbers (except for 0, but if a and b are both 0, then the statement is obviously true). So, that means we are allowed to divide by a and b.

Let b = ar and a = bq

so substitute ar for b in the second equation:
a = bq
a = arq
rq = 1
But r and q are integers, so the only way for rq to be 1 is if r and q are either both 1 or both -1.
Since a = bq and b = ar, that means either a = b or a = -b
hence gcd(a,b)>0 as a and b >0 as they both divide each other.