Let a b be nonzero integers Prove that gcda 6 1 if and only

Let a, b be nonzero integers. Prove that gcd(a, 6) = 1 if and only if gcd(a + b, ab) = 1.

Solution

Let us assume that p divides ab

Since p divides the product ab, then it must be the statement that either p divides a or p divides b

Without loss of generality, let us assume that

p divides a and if we assume that p divides (a+b), then it implies that p must also divide b

Since in our case it is given that gcd(a+b,ab) = 1

It will implies that p is either dividing a or b, since the gcd is equal to 1

Hence p divides a but it doesn\'t divide b

Hence we can say that there doesn\'t exists any common factor p between a and b

Hence gcd(a,b) = 1