CAT 2023

Slot 3

QA

Question & Solution

We know that the number of factors of these two numbers is 15. We know that the factors of 15 are 1, 3, 5, and 15.

The number of factors of N is $(p+1)\cdot(q+1)$(Where, $N=a^p\cdot b^q,$ and a, b are prime numbers).

Hence, the value of N will be least when (p+1) and (q+1) are as close as possible and a, and b are the least distinct prime numbers.

Hence, p+1 = 3 => p = 2, and q+1 = 5 => q = 4, and the prime numbers a, and b are 2, and 3, respectively.

Hence, the lowest value of N is $N=2^4\times\ 3^2\ =144$, and the second lowest value of N is $N=2^2\times\ 3^4\ =324$.

Hence, the sum is (144+324) = 468