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

Probably its an option based question. We have to come to conclusion based on the option.

Clearly,
HCF * LCM = a*b;
Where a and b are the nos.

Here HCF = 23.
And LCM = 23*13*14.

So a*b = HCF * LCM.
= 23*(23*13*14).

Now we could separate the product in two ways:

Method 1:
---------
a*b = (23*13) * (23*14).
= 299 * 322.
Hence a = 299, b =322.

Method 2:
---------
a*b = 23*23*13*14.
= 23*23*13*2*7.
= 23*23*26*7.
= (23*26)*(23*7).
Hence a = 26 and b = 7.

HCF and LCM in both the methods 1 and 2 are same.

So we have to select any one pair of assumptions based on the options given in the question.

Please explain the basics of hcf and lcm?

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.

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

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.