If d and m are the G.C.D and L.C.M of two positive integers a,b respectively , then prove that dm=ab

May 2023 1 4 Report

If d and m are the G.C.D and L.C.M of two positive integers a,b respectively , then prove that
dm=ab

Answer:

Given,

d= g.c.d(a,b)

m=L.c.m(a,b)

Since d= g.c.d(a,b)

=> a= d*x for some x

b= d*y for some y

x,y are co-primes by G.C.D definition

By definition of L.C.M,

L.c.m(a,b) is divisible by a=d*x and b=d*y

=> L..C.M(a,b) = d*x*y

=>L.C.M(a,b) × G.C.D(a,b) = d²xy

=> d×m= (d*x)(d*y)

:. dm=ab

Hence proved.