Divisibility Rules

Date: 4/10/96 at 15:4:2
From: Anonymous
Subject: Divisibility Rules of 12, Beyond!

Hello, Dr. Math! 

Here is my question: We have read divisibility rules of up to 11.  
What are the rules beyond that number? 

Brian Stelter

Date: 4/27/96 at 19:29:29
From: Doctor Steven
Subject: Re: Divisibility Rules of 12, Beyond!

Well, for 12 just check for divisibility by 3 and 4.

For 13:

1 = 1 (mod 13)

10 = -3 (mod 13)          (i.e., 10 - -3 is divisible by 13)

100 = -4 (mod 13)         (i.e., 100 - -4 is divisible by 13)

1000 = -1 (mod 13)        (i.e., 1000 - -1 is divisible by 13)

10000 = 3 (mod 13)

100000 = 4 (mod 13)

1000000 = 1 (mod 13)

So call the ones digit a, the tens digit b, the hundreds digit c, .....
and get:

a - 3*b - 4*c - d + 3*e + 4*f + g - .....

If this number is divisible by 13, then so is the original number.

You can keep doing this technique to get other formulas for 
divisibility for prime numbers.  For composite numbers 
just check for divisibility by its divisors.

A summary of divisibility rules from 3 to 13, with explanations 
for why they work, can be found at

  http://mathforum.org/k12/mathtips/division.tips.html   

Hope this helps.

-Doctor Steven,  The Math Forum