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

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

Therefore b = 23*14 is the greatest.

We know,
LCM*HCF = N1*N2.

Given,
HCF = 23 & LCM= 23*13*14.

Now,
N1*N2= (23*X) * (23*Y).
23*23*13*14= 23*23*X*Y.
X*Y=14*13.

So, N1= 23*14= 322.

Let us take two numbers as A and B.

Their HCF is 23(prime).
And their LCM is 13 and 14.

Every number is the resultant product of their HCF and LCM.SO,consider a*b=23*(13*14)
(Observe the question it's given as their LCM and their HCF).

Then a = 23*13.
And b = 23*14.

Good explanation @Lahari.

Very helpful, thanks for explaining @Lahari.

HCF of 2 numbers equal to the highest of all the common factors of the two numbers
23 is the common factor of the two numbers.

Let 23 x,23 Y be the two numbers.
we know that the factor of any two numbers is also a factor of the LCM of those two numbers.
So 23 is a factor of the two numbers 23 x and 23 y.
LCM of two numbers.
23 x and 23 Y = 13* 14 *23.

LCM =13*14 *23,
And HCF = 23,
We know that,
LCM * HCF = product of the two numbers,
13 *14 * 23 =23 x * 23 y.
@x*y= 13* 14.

Which shows that larger number will be 14 * 23 that is equal to 322.

Number 1 = a * 23
number 2 = b * 23
And other factors of LCM are 13 and 14
So clearly a * b = 13 * 14

But my point is:
a * b = 26 * 7
a * b = 81 * 2
As they are co prime in themselves,
So why is the answer not 23*81? Please explain.

Thank you for explaining the answer @Krishna.

Good explanation, thank you all.

Could not understand the concept, please help.