# 7-test and 13-test for correcting additions and multiplications

Design a seven-test and a thirteen-test for checking the correctness of additions and multiplications, based on modulo 7 and 13. Comment on the usefulness of your test.

I know that we can check that a number is divisible by 7 and 13, this is useful because these numbers are prime numbers. I could also find this website http://www.savory.de/maths1.htm where the author lays down some nice rules for divisibility tests. In class we treated the 9-test, which corresponded to adding up the digits and checking if this is divisible by $$9$$. I am not sure how we could define a useful test for 7 and 13 based on multiplications and additions.

Can somebody provide some insight?

The complexity of the divisibility test for $$d$$ depends on the length of the period of the repeating part of the fraction $$10/d$$. That depends in turn on the sequence of remainders when you divide $$10^k$$ by $$d$$ (so modular arithmetic comes into play).
That's why the tests for $$2$$, $$3$$, $$5$$ and $$9$$ are nice. The tests for $$7$$ and $$13$$ are not.
The test for divisibility by $$11$$ is fun. It uses the alternating sum of the digits. See if you can understand why.
There's a nice divisibility test for $$d=1001 = 7 \times 11 \times 13$$ that can shorten the tests for $$7$$ and $$13$$.