Modular Arithemtic Definitions by Karl Mercer

If n is a positive integer, then any two integers a and b are said to be congruent mod (modulo) n iff (if and only if) the difference between a and b is divisible by n. That is: a - b = kn, where k is any integer. This is given in the form:

In congruence mod n, where n > 0 and a, b, c, and d are any integers, the following properties hold:

Casting Out Nines

Since,

(a + b) mod n = a mod n + b mod n
(ac) mod n = a mod n * c mod n

With modulo n we know that in

where b/a, that b/9 gives us a remainder of a.

In order to find the remainder of some number divided by 9, all we have to do is count up the digits and find what the sum mod 9 is congruent to. Since we are dividing by 9, we can cancel out all the 9's or the digits that add up to nine and find the sum of the remaining digits mod 9.

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