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

Question

Accepted Answer

To solve this problem, we need to find the first two natural numbers that have exactly 15 factors.

1. **Factorization Rule**:  
   If a number $$ N $$ can be expressed as $$ N = p^a q^b $$ (where $$ p $$ and $$ q $$ are primes), the total number of factors of $$ N $$ is given by:  
   $$
   (a + 1)(b + 1).
   $$
   For a number to have 15 factors, one possible factorization is:  
   $$
   (a + 1)(b + 1) = 15.
   $$
   Possible pairs are:  
   - $$ a + 1 = 15, b + 1 = 1 \implies a = 14, b = 0 $$.  
   - $$ a + 1 = 5, b + 1 = 3 \implies a = 4, b = 2 $$.

2. **Smallest Numbers**:  
   - For $$ N = p^{14} $$: Choose  p = 2 implies N = $2^{14}$ = 16,384.  
   - For $$ N = p^4 q^2 $$: Choose  p = 2, q = 3 implies N = 2^4 * 3^2 = 144.

The next smallest number for this form is $$ N = 3^4 	imes 2^2 = 324. $$

3. **Sum of First Two Numbers**:  
   The first two smallest numbers are:  
   $$ N_1 = 144, N_2 = 324. $$
   Their sum is:  
   $$
   N_1 + N_2 = 144 + 324 = 468.
   $$

Thus, the sum of the first two natural numbers with exactly 15 factors is **468**.