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

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.

1. We know that the HCF of a prime number (when it same i.e. 23 and 23) so the HCF of 23, and 23 is 23 ok.

2. Then in question there is no LCM but here we know that the LCM will be the same as HCF when we are talking about the LCM of 23 and 23. So LCM is 23.

3. But there is a problem that 23 can not be divided by 13 and 14, hence they are not the factors of 23.

But I say they are not the factors but we make 13 and 14 factors of 23 by multiplying them with 23 ok so let's do it. 23'*13=299, 23'-14=322. Then check 13 and 14 completely divided 299 and 322.

Well Said, Thanks @Lahari.

Answer: Larger Number can be 23*14*13)

The greatest possibel number is 26*23 that satisfies the conditions. So the given answer is correct.

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.

E. G.consider the 2 number are 10 & 15
HCF of these two number is 5.(which is given)
10/5=2 and 15/5=3.

Here 2 and 3 are the least common multiples of 10 and 15.

So, if we have to find the number we will multiply the HCF which is 5 with the Least common Multiples which are 2 and 3. So that we can get the numbers 10 and 15. In that, the highest number is 15..

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.

I will explain to you with Example.
Take two numbers 21 & 36.
21 - 7 *3 & 36 - 12 *3.
H.C.F. - 3.
They gave H.C.F. and give two factors of L.C.M. 7 and 12.
Okey now simply apply logic which is greater.

Here, it is given that HCF of the two numbers are 23.

Similarly factors of LCM are 13 & 14,
Le the numbers be a & b.
We know LCM = 13*a or LCM =14*b (vice versa).
We also know LCM * HCF = a*b.
Here, 1) LCM = 13*a & HCF =23.
So, (13*a) * 23= a*b.
implies, b=23*13!
2) LCM =14*b & HCF =23.
So, (14*b) * 23 =a*b.
Implies a =23*14!
Thus, the Highest number is a= 23*14=322.