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

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.

Very helpful, thanks for explaining @Lahari.

Good explanation @Lahari.

Well explained @Krishna.

Thank you for explaining the answer @Krishna.

Thanks alot. @Dheeraj.

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 any integers.

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

@Ahmed.

Yeah, right, good explanation. Thanks.

@Chiya.

Good explained. Thanks.

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.