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

- Remove the last digit from the number.
- Double that digit.
- Subtract it from the remaining number.
- 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

1234~~5~~ 1. remove last digit
- 10 2. double digit (5)
====== 3. subtract
122~~4~~ 1. remove last digit
- 8 2. double digit (4)
===== 3. subtract
11~~4~~ 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

1477~~0~~ 1. remove last digit
- 0 2. double digit (0)
====== 3. subtract
147~~7~~ 1. remove last digit
- 14 2. double digit (7)
===== 3. subtract
13~~3~~ 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:

1~~4~~
- 8
====
-7

-7 is divisible by seven, therefore 14 is divisible by seven.

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

- subtract 21 times a constant
- 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.