Question

How to verify lcm and hcfof three numbers

Answer 1

I decided to write my comment as an answer. Rather than start with naming HCF(p,q)HCF(p,q), HCF(q,r)HCF(q,r) and HCF(r,p)HCF(r,p), start with HCF(p,q,r)HCF(p,q,r). So let's call HCF(p,q,r)=hHCF(p,q,r)=h.

Next, write HCF(p,q)=xhHCF(p,q)=xh, HCF(q,r)=yhHCF(q,r)=yh and HCF(r,p)=zhHCF(r,p)=zh. It should be clear why we can assume the factor hh appears in all three, but you also know that x,y,zx,y,z are relatively prime. (Why?)

Thus, you can write p=p′xzhp=p′xzh for some p′p′, and similarly q=q′xyhq=q′xyh and r=r′yzhr=r′yzh (again, why?). What do you get when you plug those into your equation