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:
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:
Here's what we get for a = 5 and n = 14:
Research Question 4If 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 |