# What's the quickest way to determine the proper divisors of a number by hand?

## For example, the proper divisors of 130 are 1, 2, 5, 10, 13, 26, and 65 - are there any patterns I could use to figure those out quickly?

Feb 11, 2017

Not much, but here are some ways to find some of them:

#### Explanation:

Let $n$ be that number (let's say it's a positive integer). Then:

$1$ and $n$ are divisors.

If $n$ is even (last digit is $2 , 4 , 6 , 8 , 0$) it is divisible by $2$ and $\frac{n}{2}$

If the sum of $n$'s digits is a multiple of $3$, it is divisible by $3$ and $\frac{n}{3}$

If the last two digits are $0$ or a multiple of $4$, it is divisible by $4$ and $\frac{n}{4}$

If the last digit is $5$ or $0$, it's divisible by $5$ and $\frac{n}{5}$

If it's divisible by $3$ and even, it's divisible by $6$ and $\frac{n}{6}$

If $\frac{n}{4}$ is even, it's divisible by $8$ and $\frac{n}{8}$

If the sum of $n$'s digits is a multiple of $9$, it's divisible by $9$ and $\frac{n}{9}$

If the last digit is $0$, it's divisible by $10$ and $\frac{n}{10}$

Rules for $3$ and $9$ can be repeated, e.g. if you arrive at a multi-digit number, say $m$, you can repeat the process to see if $m$ is divisible by $3$ or $9$ respectively, then if it is, $n$ will also obey the third and eighth rules.

Hope this helps.