To Prove the relation between HCF and LCM of three numbers if $p,q,r$ are three positive integers prove that
$$LCM(p,q,r)=\frac{pqr \times HCF(p,q,r)}{HCF(p,q) \times HCF(q,r) \times HCF(r,p)}$$
I tried in this way;
Let $HCF(p,q)=x$ hence $p=xm$ and $q=xn$ where $m$ and $n$ are relatively prime.
similarly let $HCF(q,r)=y$ hence $q=ym_1$ and $r=yn_1$ where $m_1$ and $n_1$ are Relatively prime.
Alo let $HCF(r,p)=z$ hence $r=zm_2$ and $p=zn_2$
we have $$p=xm=zn_2$$
$$q=xn=ym_1$$and
$$r=yn_1=zm_2$$
can i have any hint to proceed?
 A: I decided to write my comment as an answer. Rather than start with naming $HCF(p,q)$, $HCF(q,r)$ and $HCF(r,p)$, start with $HCF(p,q,r)$. So let's call $HCF(p,q,r) = h$.
Next, write $HCF(p,q) = xh$, $HCF(q,r) = yh$ and $HCF(r,p) = zh$. It should be clear why we can assume the factor $h$ appears in all three, but you also know that $x,y,z$ are relatively prime. (Why?)
Thus, you can write $p = p'xzh$ for some $p'$, and similarly $q = q'xyh$ and $r = r'yzh$ (again, why?). What do you get when you plug those into your equation?
A: To get a start on the problem first let us try to understand it intuitively. We need to show that for three numbers $a,b,c$, $$abc=\frac{LCM(a,b,c) \times HCF(a,b) \times HCF(b,c) \times HCF(c,a)}{HCF(a,b,c)}$$
On the LHS- the product of three numbers $a, b, c$ consists of product of all their factors.
On the RHS- The LCM contains all the factors of $a,b,c$ with common factors only taken once. The pairwise HCF's get us the pair-wise common factors not accounted for by the LCM. However, now the factors common to all the three numbers have been counted 4 times (with the LCM as well as with each pair-wise HCF's), instead of being counted 3 times (once for each number). To fix this  we divide the entire product by $HCF(a,b,c)$ which gets us the factors common to all the three numbers.
We can construct a general proof of the theorem based on above argument as well.
Alternatively, we can see that of the three number $a,b,c$ any number number, for example, $a$ is made of 4 kind of factors-
-Factors only in $a$ (say product of these factors is $\alpha$)
-Factors common between $a,b$ (say product of these factors is $x$)
-Factors common between $a,c$ (say product of these factors is $y$)
-Factors common between all three numbers (say product of these factors is $w = HFC(a,b,c)$).
Thus
$a = \alpha.x.y.w$
Similarly,
$b = \beta.x.z.w$ and
$c = \gamma.y.z.w$
Therefore, by multiplying the equations for three numbers we get,
$abc = \alpha.\beta.\gamma.x^2.y^2.z^2.w^3.$
or, $abc = \frac{\alpha.\beta.\gamma.x^2y^2.z^2.w^4}{w}$
but, $x.w=HCF(a,b)$ and $y.w=HCF(b,c)$ and $z.w=HCF(c,a)$ and $w=HCF(a,b,c)$
therefore,
$abc = \frac{\alpha.\beta.\gamma.x.y.z.w.HCF(a,b).HCF(b,c).HCF(c,a)}{HCF(a,b,c)}$
finally, we also have that $LCM(a,b,c)=\alpha.\beta.\gamma.x.y.z.w$
thus, $$abc = \frac{LCM(a,b,c).HCF(a,b).HCF(b,c).HCF(c,a)}{HCF(a,b,c)}$$
A: Let $P, Q$ and $R$ be the sets of prime factors of the numbers $p,q$ and $r$ respectively. Let the universal set be $P\cup Q\cup R$. Let there be product function $v(S)$ which gives the product of the elements of set S.
Then, celarly,
$$pqr=v(P\cap Q' \cap R')*v(Q\cap P' \cap R')*v(R\cap P'\cap Q')*v^2(P\cap Q\cap R')*v^2(Q\cap R\cap P')*v^2(R\cap P\cap Q')*v^3(P\cap Q\cap R)$$
Now,
$$LCM(p,q,r)=v(P\cap Q' \cap R')*v(Q\cap P' \cap R')*v(R\cap P'\cap Q')*v(P\cap Q\cap R')*v(Q\cap R\cap P')*v(R\cap P\cap Q')*v(P\cap Q\cap R)$$
$$HCF(p,q)=v(P\cap Q\cap R')*v(P\cap Q\cap R)$$
$$HCF(q,r)=v(Q\cap R\cap P')*v(P\cap Q\cap R)$$
$$HCF(r,p)=v(R\cap P\cap Q')*v(P\cap Q\cap R)$$
$$HCF(p,q,r)=v(P\cap Q\cap R)$$
These expressions satisfy your equation.
