Divisibility by Seven

By: Su, Francis E., et al. "Divisibility by Seven." Mudd Math Fun Facts Knowing divisibility rules is an important bit of mathematical / numerical knowledge. In grade school, everyone learns some simple tests for divisibility by small numbers such as 2, 3, 5, and 9. But far less well-known are some simple divisibility tests for the number 7. Here are a couple: Test #1. Take the digits of the number in reverse order, from right to left, multiplying them successively by the digits 1, 3, 2, 6, 4, 5, repeating with this sequence of multipliers as long as necessary. Add the products. 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. Remove the last digit, double it, subtract it from the truncated original number and continue doing this until only one digit remains. 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.