Aptitude - Problems on H.C.F and L.C.M - Discussion

Please explain the basics of hcf and lcm?

HCF is common factor while LCM is greatest among the given numbers.

L.C.M of two Numbers x1 = 13 & x2 = 14.
H.C.F (highest common factor) of x1 & x2 is = 23.

Therefore x1 = 23 & x2 = 23.

Largest Number is,

L.C.M(x1)*H.C.F(x1) = 13*23 = 299.
L.C.M(x2)*H.C.F(x2) = 14*23 = 322.

So answer = 322.

First we have to understand the meaning of factor.

i.e. If number a is divided another number b exactly we say that a is factor of b.

Hence as per this factors are 13 and 14. So to find largest number.
Let us consider x and y are numbers.

x/factor = HCF.
y/factor = HCF.

x/13 = 23, x = 13*23 x = 299.
y/14 = 23, y = 14*23 y = 322.

So largest number is 322. Simple and less time consumption.

We mean (Number = H.C.F*L.C.M) it's wrong or not please explain me.

We know that product of two numbers = H.C.F*L.C.M.

X*Y = H.C.F*L.C.M.

In the question they gave H.C.F of two numbers is 23. And the factors of their L.C.M are 13 and 14.

X*Y = H.C.F*L.C.M | check | L.C.M factors.

X*Y = 23*(13*14) |of two no's.

So |X = 23*13 = 299; |23|322, 299 ||14, 13.

Y = 23*14 = 322; ||.

So the greatest among the two no's is 322 |.

The L.C.M Of 2 numbers are (23x13) (23x14).

23x13 = 299, 23x14 = 322, which is largest number = 322.

I just want to know how these 13 and 14 obtained?

Let 'a' and 'b' be the two numbers.

HCF (a, b) = 23.

LCM (a, b) = 13*a = 14*b (or) LCM = 14*a = 13*b.

Note : The (or) case is just to indicate that we could have taken the factors the other way round ---> since they are factors for LCM. Both cases would result in the same large number.

Theorem: The product of two numbers = a*b = LCM (a, b)*HCF (a, b).

Applying the above theorem here :

=> a*b = HCF (a, b)*LCM (a, b).

=> a*b = 23*(13*a) (or).

Therefore => b = 23*13.

Similarly, a = 23*14.

The largest of the 2 numbers = 23*14 = 322 is the answer.

Read the question clearly.

Given that the two factors is 13 and 14 this means 23 is also a factor of L.C.M.

So LCM is 23*13*14.

Now 23 is HCF so suppose two numbers are 23*x and 23*y.

By theorem; (23*x)(23*y) = (23)(23*13*14).

This gives x equal to either 13 or 14 and similarly y.