division rule for seven

The easiest division rule is for two: if the last digit's even, then two divides the number. There's a rule for five: if the last digit is a five or a zero, then five divides the number. Three's a little trickier: add up the digits, and if they are divisible by three then three divides the number. There's a simple division rule for eleven, too.

But a division rule for seven?

Yes, it exists. The rule's not quite as easy as summing the digits, but it's simple enough that you don't have to break out long division.

Here's what you do

1. Remove the last digit from the number.
2. Double that digit.
3. Subtract it from the remaining number.
4. If the number you've got is divisible by seven, the original number is divisible by seven. Keep going until you get to a small number.

The small number you end up with can be positive or negative, like "0", "3", "7", "14", "-7". The only thing that matters is whether seven divides the number. (Seven does divide "-7" and "0" as well as the more traditional "7", "14", ...)

Example 1

  12345     1. remove last digit
-   10      2. double digit (5)
======      3. subtract
  1224      1. remove last digit
-   8       2. double digit (4)
=====       3. subtract
  114       1. remove last digit
-  8        2. double digit (4)
====        3. subtract
   3

3 is not divisible by seven, therefore 12345 is not divisible by seven.

Example 2

  14770     1. remove last digit
-    0      2. double digit (0)
======      3. subtract
  1477      1. remove last digit
-  14       2. double digit (7)
=====       3. subtract
  133       1. remove last digit (3)
-  6        2. double digit (3)
====        3. subtract
   7

7 is divisible by seven, therefore 14770 is divisible by seven.

This works even for simple numbers:

   14
-  8
====
  -7

How does it work?

In mathematical terms, you can think of what you're doing as

1. subtract 21 times a constant
2. divide evenly by 10

You can subtract 21 times a constant without affecting divisibility, because 21 is divisible by 7, so you're just subtracting multiples. You can divide evenly by 10, because 10 is relatively prime to 7: you're never subtracting factors.