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.