Natural Numbers with 15 Factors

The number of factors of $N = p_1^{a} p_2^{b} \dots$ is given by $(a+1)(b+1)\dots = 15$.

Since $15 = 15 \times 1$ or $5 \times 3$, a number with 15 factors must be of the form:

1. $p^{14}$
2. $p^4 q^2$ (where $p, q$ are distinct primes)

Let's test small prime values for $p^4 q^2$:

• $p = 2, q = 3 \implies 2^4 \times 3^2 = 16 \times 9 = 144$
• $p = 3, q = 2 \implies 3^4 \times 2^2 = 81 \times 4 = 324$
• $p = 2, q = 5 \implies 2^4 \times 5^2 = 16 \times 25 = 400$

For $p^{14}$, the smallest is $2^{14} = 16384$, which is much larger.

Thus, the first two natural numbers having 15 factors are $144$ and $324$.

Their sum = $144 + 324 = 468$.