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

Given an integer

aand a positive integern, which integersmsatisfyma0 (modn)?The

additive orderofamodulonis defined to be the smallest positive integermthat satisfies the congruence equationma0 (modn). 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 computema%nwithn= 10,a= 6, and values ofmbetween 1 and 20:

As we can see, for these values of

mwe havema0 (modn) form= 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 ofaandnas input, computesma%nwith lots of integersm, and then makes a list of those values ofmsuch thatma0 (modn) Here it is in action using the values ofaandnfrom above:Here's what we get for

a= 5 andn= 14:

## Research Question 4

If we keep

nfixed and replaceaby another integerbcongruent toa(modn), how will the output from the preceding applet change?

## Research Question 5

Find the form of all values of

mthat satisfyma0 (modn).

Section 3.1 | Section 3.2 | Section 3.3 | Section 3.4

Copyright © 2001 by W. H. Freeman and Company