Understanding the Number of Factors
Example 1: Finding Factors of 8
To understand the number of factors for any given number, let's take the example of number 8. The factors of the number 8 are: 1, 2, 4, 8.
So let's analyze how these factors are coming. If we prime factorize this number 8, it will be $2^3$.
Now every time we take a power of 2, we create a factor:
- $2^0 = 1$ is a factor
- $2^1 = 2$ is a factor
- $2^2 = 4$ is a factor
- $2^3 = 8$ is a factor
So out of the 3 powers of 2, we are simply selecting either 0, 1, 2, or 3. So the way to do it is through a total of 4 ways, i.e., to pick one of (0, 1, 2, 3).
Example 2: Finding Factors of 12
Let's take another number, for example 12. So 12 can be written as $2^2 \times 3^1$.
Now a factor of 12 can be created by borrowing some/none power of 2 and some/none power of 3 (included none since we don't take any power of 2 or 3, it will simply create 1 which in itself would be a factor as well).
$2^0 \times 3^0 = 1$
So, factors of 12 would be:
- $2^0 \times 3^0 = 1$
- $2^1 \times 3^0 = 2$
- $2^0 \times 3^1 = 3$
- $2^2 \times 3^0 = 4$
- $2^1 \times 3^1 = 6$
- $2^2 \times 3^1 = 12$
What we are doing here is borrowing powers of 2 in $(2+1) = 3$ ways and borrowing powers of 3 in $(1+1) = 2$ ways, so a total of $3 \times 2 = 6$ ways. That is what is giving us the number of factors.
General Formula
So now generalizing for any number $N$, when prime factorized looks like:
$$N = 2^p \times 3^q \times 5^r \times \ldots$$
Then the total number of factors for that number $N$ would be $(p+1)(q+1)(r+1)\ldots$
Solving the Given Problem
So here in this question it is given that the number of factors is 15 and we have to find the sum of first 2 such numbers.
Now 15 can be written as $3 \times 5$.
So by the formula of number of factors it should be: $$(p+1)(q+1) = 15$$
So $p = 2$ and $q = 4$ comes from this equation.
Since asked in the question about the first such natural numbers, so we try to create the smallest such number that can be made.
From this finding we can say that the exponents in the prime factorization of that number are 2 and 4.
Hence the 2 smallest numbers that can be made are:
- $2^4 \times 3^2 = 16 \times 9 = 144$
- $3^4 \times 2^2 = 81 \times 4 = 324$
Hence the asked sum would be $144 + 324 = 468$.