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.

@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.

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.

Given: HCF of A and B is 23.

Conclusion: Since 23 is the greatest common factor, both A and B must be multiples of 23.

Let A = 23 * x, where x is some integer.
Let B = 23 * y, where y is some integer.

Note: x and y don't have any common factors other than 1 (otherwise, the HCF would be greater than 23).

Finding the LCM:

LCM is the smallest number that is a multiple of both A and B.

A is a multiple of 23 and x.
B is a multiple of 23 and y.
To include all factors of both A and B, we need to include 23, x, and y in the LCM. Since x and y don't share any common factors, we simply multiply them all together.

Therefore, LCM(A, B) = 23 * x * y.
If x and y had a common factor other than 1, it would mean that the HCF of A and B would be greater than 23.
Here's why:
Let's say x and y share a common factor of 'd'.
So, x = d * p and y = d * q, where p and q don't have any common factors.
Now, A = 23 * x = 23 * d * p and B = 23 * y = 23 * d * q.
In this case, the HCF of A and B would be 23 * d, which is greater than 23.
This contradicts our initial assumption that the HCF is 23.
Therefore, for the formula LCM(A, B) = 23 * x * y to hold, x and y must not have any common factors other than 1.

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.

Very nice explanation. Thanks all.

I am not getting this, Can anyone explain me more clearly?

Good explanation, thanks.

Very helpful. Thanks all for giving the description.

Good explanation. Thanks a lot @Hari Bol.