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?

1 Answer
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 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.