Question

Prove that LCM (a,b,c)=abc\ HCF (ab,bc,ca)

Answer 1

Given:

3 Numbers a. b and c.

To Prove:

LCM (a,b,c)=abc\ HCF (ab,bc,ca)

Solution:

We have three numbers a , b , c .

Let HCF be the highest common factor of a, b and c.

• a = HCF x r
• b = HCF x s
• c = HCF x t

Where HCF(r,s,t) = 1 since HCF(a,b,c) = HCF

• Here HCF x r x s x t = LCM(a,b,c)
• rstxHCF = LCM(a,b,c)

Now,

• a x st = HCF x rst
• b x rt = HCF x rst
• c x rs = HCF x rst

Multiply all three equations,

• abc x (rst)² = HCF³ x (rst)³
• abc = HCF x rst x HCF²
• abc = LCM x HCF²

Now HCF(ab,bc,ca) :

• ab = HCF x r x HCF x s = HCF² x rs
• bc = HCF x s x HCF x t = HCF² x st
• ac = HCF x r x HCF x t = HCF² x rt

Therefore HCF(ab,bc,ca) = HCF²

Therefore ,

• abc = LCM(a,b,c) x HCF(ab,bc,ca)
• LCM(a,b,c) = abc/HCF(ab,bc,ca)

Thus proved that LCM (a,b,c)=abc\ HCF (ab,bc,ca).