Number Properties : LCM, HCF

This question is a relatively easy question on LCM and HCF of two numbers. 

Question

If the product of two positive integers is 144, which of the following could be the LCM and HCF of the two numbers?
I. LCM : 24; HCF : 6
II. LCM : 18; HCF : 8
III. LCM : 16; HCF : 9

A. I only
B. II and III only
C. I and II only
D. I and III only
E. I, II and III

Correct Answer : Choice A

Explanatory Answer

Result 1:
The product of two positive integers 'a' and 'b' is equal to the product of the LCM (a, b) and HCF (a, b).
i.e., a * b = LCM (a, b) * HCF (a, b)

Result 2:
HCF is a factor of both the positive integers.
LCM is a multiple of both the positive integers.
So, it is evident that the LCM of the two positive integers has to be a multiple of the HCF of the two numbers.

Combining these two results, we have to find out which among the pairs has a product of 144 such that the LCM is a multiple of the HCF.

I. LCM : 24 and HCF : 6. Product of the LCM and HCF = 24 * 6 = 144. The LCM 24 is a multiple of the HCF 6. Hence, this is a possible pair.

II. LCM : 18 and HCF : 8. Product of the LCM and HCF = 18 * 8 = 144. However, the LCM 18 is NOT a multiple of the HCF 8. Hence, this one is not a possible pair.

III. LCM : 16 and HCF : 9. Product of the LCM and HCF = 16 * 9 = 144. However, the LCM 16 is NOT a multiple of the HCF 9. Hence, this one is also not a possible pair.