Number of factors

There’s a neat trick for finding the number of factors of a number. First find the prime factorization of the number, for example:

$756 = 2^2 \cdot 3^3 \cdot 7^1$

Add $+1$ to the power of each prime factor and then multiply those numbers together. In this example,

$(2+1)\cdot(3+1)\cdot(1+1)= 3\cdot 4 \cdot 2 = 24$ factors.

This reflects the fact that each factor of $756$ contains $2^n$ where $n= 0, 1$ , or $2$ and $3^m$ , where $m = 0, 1, 2$ , or $3$ and $7^p$ where $p = 0$ or $1$ .

I used this property to solve a problem in a number theory class I’m taking. The problem asks to compute the sum of all positive integers $k$ such that $1984k$ has $21$ positive factors.

Since $21 = 3\cdot7$ , working backward we are looking for a prime factorization with 2 primes raised to the powers of 2 and 6. The prime factorization of $1984 = 2^6\cdot 31$ so we have one prime raised to the sixth power. All we need is the other prime to be squared. Setting $k=31$ gives us