Aptitude - Problems on H.C.F and L.C.M - Discussion
Discussion Forum : Problems on H.C.F and L.C.M - General Questions (Q.No. 2)
2.
The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:
Answer: Option
Explanation:
Clearly, the numbers are (23 x 13) and (23 x 14).
Larger number = (23 x 14) = 322.
Discussion:
153 comments Page 16 of 16.
Subham mandal said:
12 months ago
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.
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.
(27)
Yohannes Tesfaye said:
4 months ago
Good explanation, thank you all.
(2)
Vadrevu sai lalitha anjani sri said:
3 months ago
I am not getting this, Can anyone explain me more clearly?
(9)
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