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(a,b). Prove the following for all positive integers a, b, and c:

a * lcm(b,c) = lcm(ab, ac).

Solution

Let, lcm(b,c)=r

So, r=kb,r=lc, gcd(k,l)=1

a*lcm(b,c)=ar

Hence,

ar=alc=l(ac), ar=k(ab)

ar=l(ac),ar=k(ab)

And , gcd(l,k)=1

Hence, ar=lcm(ab,ac)

HEnce proved