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?