Question

Find the sum of the two smallest natural numbers with the number of factors as 15.

Answer 1

Let's consider a number with prime factorization pa ⋅ qb , where p and q are distinct prime numbers, and a and b are positive integers. The number of factors (N) of this number is given by (a +1)⋅(b +1).

For a number with 15 factors, we can have either a =14 and b =0 or a =2 and b =4 (or vice versa).

1. If a =14 and b =0, the number is p 14.
2. If a =2 and b =4, the number is 2⋅ p 2⋅ q 4.

Now, we need to find the two smallest natural numbers with these factorizations.

1. For a =14 and b =0, the smallest number is 214=16,384214=16,384.
2. For a =2 and b =4, the smallest number is 22⋅34=14422⋅34=144.

The sum of the two smallest natural numbers with 15 factors is 16,384+144=16,52816,384+144=16,528.

Therefore, the answer is 16,528.