The positive least common multiple of a and b is denoted by

The positive least common multiple of a and b is denoted by lcm(ab,b). Prove the following for all positive integers a, b, c:

If a=a1*c and b=b1*c, where c=gcd(a,b), then lcm(a,b)=a1*b1*c.

(Use prime factorization by showing each side is equal)

Solution

If a = a1 * c and

b = b1 * c it implies that a1 and b1 are prime numbers

As it is given c = gcd( a , b )

Therefore , lcm( a , b ) = a1 * b1 * c