A basic question in the study of congruences is the following:

Given an integer a and a positive integer n, which integers m satisfy

ma 0 (mod n)?

The additive order of a modulo n is defined to be the smallest positive integer m that satisfies the congruence equation ma 0 (mod n). In order to get a feel for the above question, what's the first thing that we do? Repeat three times: "Try some examples." Here's what we get if we compute ma % n with n = 10, a = 6, and values of m between 1 and 20:

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As we can see, for these values of m we have ma 0 (mod n) for m = 5, 10, 15, and 20. (Thus the additive order of 6 modulo 10 is equal to 5.) It is also clear that there is a bunch of extra information in the above output that we don't need. The applet below provides "just the facts, ma'am." It takes specific values of a and n as input, computes ma % n with lots of integers m, and then makes a list of those values of m such that ma 0 (mod n) Here it is in action using the values of a and n from above:

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Here's what we get for a = 5 and n = 14:

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#### Research Question 4

If we keep n fixed and replace a by another integer b congruent to a (mod n), how will the output from the preceding applet change?

#### Research Question 5

Find the form of all values of m that satisfy ma 0 (mod n).

Section 3.1 | Section 3.2 | Section 3.3 | Section 3.4