Question

Three numbers are in the ratio of 2 : 3 : 4 and their L.C.M. is 240. Their H.C.F. is:

Answer 1

$2x + 3x + 4x = 240 \\ 9x = 240 \\ x = 240 \div 9$

Answer 2

Given ratio = 2 : 3 : 4.

Let the three numbers = 2x,3x and 4x.

Now, we have to find the LCM of these three numbers.

⇒ Prime factorization of 2 = 2

⇒ Prime factorization of 3 = 3

⇒ Prime factorization of 4 = 2 * 2

Now,

LCM(2,3,4)x = (2 * 2 * 3)x

                    = 12x.

But,

Given LCM = 240

⇒ 12x = 240

⇒ x = 240/12

⇒ x = 20.

Then,

⇒ 2x = 40

⇒ 3x = 60

⇒ 4x = 80.

Now,

⇒ Prime factorization of 40 = 2^3 * 5

⇒ Prime factorization of 60 = 2^2 * 3 * 5

⇒ Prime factorization of 80 = 2^4 * 5

HCF(40,60,80) = 2^2 * 5

                         = 20.

Therefore, the HCF of three numbers is 20.

Hope it helps!