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

Well explained @Krishna.

Good explanation @Lahari.

Very helpful, thanks for explaining @Lahari.

Please correct me, if I am wrong:

Let two numbers be A and B. It is given that their HCF is 23.

Therefore, A = 23 * x
and B = 23 * y, where x and y are coprime pairs, so according to the property of coprime LCM(x,y) = xy.

And we can find their LCM as follows: LCM(A, B) =LCM(23*x, 23*y) = 23 * x * y.

Therefore, x and y are the other two factors of LCM, and we are given that those 2 factors are 13 and 14 respectively.

Therefore x = 13 and y = 14.

And hence largest number = 23 * 14 = 322.

@All.

This question has a very clear concept of the fundamental theorem of arithmetic
Let one no. be =23a and the second no. =23b.

According to question;

First num. * Second num = hcf * lcm.
23a * 23b = 23 * lcm.

(Note the concept that HCF is always a factor of lcm )
So we can write;

23a* 23b= 23* 23* 13 * 14 ( as given in the question that 13 and 14 are the other factors )
so, ab = 13*14.

Now ACCORDING TO the FUNDAMENTAL THEOREM OF ARITHMETIC;

We need to express ab as the product of unique prime factors
ab = 13*7*2
Now, there can be 3 possible conditions
a can be
1. 13*7*2 ( then b will be 1) { note= you can do the same thing with b also there will be no change, only the last answer will be a=322}
2. 13*7 (then b will be 2)
3. 13 ( then b will be 14).

According to all the 3 conditions, the two numbers can be-
FOR 1nd condition.
first no. - 4186 ( 23a= 23 *13*7*2)
second no. - 23 (23b=23*1).

FOR 2nd condition.
First no. - 2743 (23a=23*13*7)
Second no. - 46 (23b = 23*2).

FOR 3 condition
First no. - 299 (23a = 23*13)
Second no. - 322 (23b=23*14)

So, if we see in the given question the options are given according to 3 condition
Hence, according to the 3rd condition, the greater number is b=322.
I hope it helps.

Good explanation. Thanks a lot @Hari Bol.

Good explanation, thanks.

Very nice explanation. Thanks all.

HCF of two numbers is the number which is the common factor for both numbers given Here 23 is the common factor.
Other than this common factor, we also will have the product of uncommon factors for the two numbers (13 and 14 here).
The first number = 23 × 13 = 299 and
secondly number =23 × 14 = 322,
The greatest of two numbers is 23×14 = 322.

Very helpful. Thanks all for giving the description.