7 (number)

• 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).[1]
3. Repeat the second step again and again until it is clearly 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.