Thread: prove gcd and lcm

Show that if a and b are positive integers, then there are divisors c of a and d of b with gcd(c,d)=1 and cd=lcm[a,b]

Originally Posted by mandy123

Show that if a and b are positive integers, then there are divisors c of a and d of b with gcd(c,d)=1 and cd=lcm[a,b]

Set c = a/gcd(a, b), this makes sure c and b have no common divisors. And with d = b, c and d have no in common either.

So, gcd(c, d) = 1.

And, since lcm(x, y) can be defined as x*y/gcd(x, y), cd=a/gcd(a, b)*b = lcm(a, b)