CAT 2023 Slot 3 QA Question 6: The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Question

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Accepted Answer

If $N = p^a q^b$, the number of factors is $(a + 1)(b + 1)$. Since 15 can be written as $15 	imes 1$ or $5 	imes 3$, the smaller numbers will come from exponents 4 and 2 on the smallest primes.

$2^4 	imes 3^2 = 16 	imes 9 = 144$
$2^2 	imes 3^4 = 4 	imes 81 = 324$

Other forms such as $2^{14}$