Question

If a=bc with hcf(b,c)=1, then lcm(a,d) = lcm(c,bd)

Answer 1

Basically, we prove, with (b,c) = 1;
lcm ( bc , d ) = lcm ( c , bd )

If (d,bc) = 1;
lcm(bc,d) = bcd = lcm (c,bd) ;

If (d,bc) = m;
=> d = mx; bc = my;
So,
lcm( bc,d ) = lcm( mx, my ) = mxy ;
lcm( c,bd ) = lcm( c, mxy/c) = mxy;

.'. lcm(a,d) = lcm(c,bd) ......... for hcf(b,c) = 1;