7 (number)

Divisibility Rule

• To test if an integer is divisible by 7 or not,

1. Take off the last digit and multiply it by 2.
2. Subtract that product from the root number (the integer without the last digit).
3. Repeat the second step again and again until it is clear that the number is divisible by 7 or not.

Example

918785

1. 5×2=10, 91878-10=91868
2. 8×2=16, 9186-16=9170
3. 0×2=0, 917-0=917
4. 7×2=14, 91-14=77

77 is divisible by 7, so the original number 918785 is also divisible by 7.

Proof: Pretend ${\displaystyle a}$ is the root number and ${\displaystyle b}$ is the last digit. Then ${\displaystyle 10a+b}$ is the whole number. If the whole number (${\displaystyle 10a+b}$) is divisible by 7, ${\displaystyle 3a+b}$ is also a multiple of 7. And ${\displaystyle (3a+b)-7b=3a-6b}$ is also a multiple 7. That proves ${\displaystyle 3a-6b}$ is divisible by seven, since 3 and 6 are both multiples of 3. So ${\displaystyle a-2b}$ is also divisible by 7.