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

The correct answer is 468.

To find the natural numbers that have exactly 15 factors, we can express a natural number in its prime factorization form:

$N=p_1^{a_1}\times p_2^{a_2}\times \cdots \times p_k^{a_k}$

where $p_1,p_2,\cdots ,p_k$ are distinct prime numbers, and $a_1,a_2,\cdots ,a_k$ are non-negative integer exponents.

The total number of factors of $N$ (including 1 and the number itself) is given by the formula:

$\text{Number of factors}=(a_1+1)(a_2+1)\cdots (a_k+1)=15$

Since 15 can be factored into positive integers greater than 1 in only two ways, we have two possible cases:

Case 1: The factorization of 15 is 15 itself

This implies there is only one prime factor:

$a_1+1=15\Rightarrow a_1=14$

So the number is of the form $N=p^{14}$. The smallest such number occurs when we choose the smallest prime $p=2$:

$N=2^{14}=16384$

Case 2: The factorization of 15 is 3 × 5

This implies there are two distinct prime factors:

$a_1+1=3\Rightarrow a_1=2$

$a_2+1=5\Rightarrow a_2=4$

So the number is of the form $N=p_1^2\times p_2^4$, where $p_1$ and $p_2$ are distinct prime numbers.

To find the smallest numbers of this form, we substitute the smallest prime numbers (2, 3, 5) into the equation:

1. If we choose $p_1=3$ and $p_2=2$:

$N=3^2\times 2^4=9\times 16=144$

2. If we choose $p_1=2$ and $p_2=3$:

$N=2^2\times 3^4=4\times 81=324$

3. If we choose $p_1=5$ and $p_2=2$:

$N=5^2\times 2^4=25\times 16=400$

Comparing the values obtained from both cases, the first two smallest natural numbers having exactly 15 factors are 144 and 324.

The sum of these two numbers is:

$144+324=468$