Divisibility by Seven

Divisibility by Seven

A simple divisibility tests for the number 7:

Test #1 -

Step 1 : Take the digits of the number in reverse order
Step 2 : From right to left, multiply them successively with the digits 1, 3, 2, 6, 4, 5
Step 3 : Repeat with this sequence of multipliers as long as necessary
Step 4 : Add the products
Voila! This sum has the same remainder mod 7 as the original number!

Example: Is 1603 divisible by seven? Well, 3(1)+0(3)+6(2)+1(6)=21 is divisible by 7, so 1603 is.


Test #2 -

Step 1 : Remove the last digit
Step 2 : Double it
Step 3 : Subtract it from the truncated original number and continue doing this until only one digit remains.
Voila! If this is 0 or 7, then the original number is divisible by 7.

Example: 1603 -> 160-2(3)=154 -> 15-2(4)=7, so 1603 is divisible by 7.

See the reference for more tests and more references.