Show that the product of the least common multiple and the greatest common divisor of two positive integers a and b equa...

Question

Show that the product of the least common multiple and the greatest common divisor of two positive integers a and b equals a * b. That is, prove: lcm(a,b) * gcd(a,b) = a * b.

Accepted Answer

lcm(a,b) * gcd(a,b) = a * b for all positive integers a, b. Every positive integer can be uniquely written as a product of prime powers. For any prime p, the exponent of p in gcd(a,b) is the smaller of its exponents in a and b, while in lcm(a,b) it is the larger. Since min(x,y) + max(x,y) = x + y for any x, y, the sum of exponents in gcd and lcm equals the sum in a and b. Multiplying over all primes gives lcm(a,b) * gcd(a,b) = a * b.