Let a b Z 0 If d gcda b and m lcma b prove that dm abSol

Let a, b Z \\ {0}. If d = gcd(a, b) and m = lcm(a, b), prove that dm = |ab|.

Solution

Let d = gcd(a,b)
Then a = d * x for some x, and b = d * y for some y.
x and y are coprime by defintion of gcd.

By definition, lcm(a,b) is divisible by a = d * x and b = d * y, therefore
m = lcm(a,b) = d * x * y (since x and y are coprime)

lcm(a,b)*gcd(a,b) = m * d = (d^2) * x * y = (c * x) * (c * y) = a * b