### 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.

Dhiraj Reddy says

Nice Question.

I think It Might Be 1 Option.