Question

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?

Answer 1

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 `n/2`

If the sum of `n`'s digits is a multiple of `3`, it is divisible by `3` and `n/3`

If the last two digits are `0` or a multiple of `4`, it is divisible by `4` and `n/4`

If the last digit is `5` or `0`, it's divisible by `5` and `n/5`

If it's divisible by `3` and even, it's divisible by `6` and `n/6`

If `n/4` is even, it's divisible by `8` and `n/8`

If the sum of `n`'s digits is a multiple of `9`, it's divisible by `9` and `n/9`

If the last digit is `0`, it's divisible by `10` and `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.