If a b are in Z prove that a b ab d where d gcdabSolution

If a, b are in Z, prove that (a) + (b) =(a,b) = (d) where d = gcd(a,b)

Solution

proof- since If a, b are in Z

gcd(a, d)gcd(b, d)
= [(a)+(d)][(b)+(d)]
= (a)(b) + (a)(d) + (d)(b) + (d)(d)
= (a)(b) + [(a)+(b)+(d)](d)
= (ab) + (Z+(d))(d)
= (ab) + Z(d)
= (ab) + (d)
= gcd(ab, d). proved