S. I. M.
Mr. Plassmann
Section 1.4: Modular Arithmetic
Objectives:
1.
Be able to identify the elements of a modular system and perform operations with elements from that
modular system.
2.
Understand how representations of numbers in our number system translate to elements of a modular system.
Vocabulary:
Modular Arithmetic
Modulus
Residue
Congruent
P. O. T. D.:
1.
Find the prime factorizations of
1265 and 550 .
Use their prime factorizations to find their LCM and
GCD.
2.
Use Euclid’s algorithm to find the GCD of
23, 647
and
1, 764 .
Lesson:
I. Modular Arithmetic
To help us gain insight and perspective to our own number system, we will investigate a number of
alternative systems of arithmetic. Some fundamental examples of systems of arithmetic are given by socalled modular arithmetics. Probably the easiest way to explain what we mean by a modular arithmetic is
to look at a simple example.
Consider the set
we divide by
7
7
  0, 1, 2, 3, 4, 5, 6 .
using the division algorithm. The number
is called the system of least non-negative residues modulo
This is the set of possible remainders we obtain if
7
7.
is called the modulus of our system, and
We are going to equip
7
7
with two
operations: addition and multiplication. To describe these operations, it is convenient to arrange the elements
of
7
in a circle as follows.
Now suppose we want to add two elements of
the above circle and proceed clockwise a total of
7.
5
Here are some more examples of addition in
1  2  3;
6  6  5;
7,
say we want to add
units. That brings us to
5
4.
to
6.
Then we start at
6
56  4
in
So we say
7.
2  5  0; 3  4  5  6  4
on
It is also possible to multiply in
just add
times:
6
to itself twice:
7.
Suppose for example, we want to multiply
6 2  6  6  5.
5 4  5555  6 .
Similarly, to multiply
5
by
4
6
by
5
we add
Here are a few more examples of multiplication in
2 3  6;
3 4  5;
4 6  3;
2.
Then we
to itself
4
7.
5 5 56
Example #1: Using Modular Arithmetic
See the attached handout
Once we understand addition and multiplication, we can try to do more interesting calculations in
To get you started on this one, let’s take a look at another example: What is
question, we must first realize that
element
x
in
7
more than one such
for which
x
in
1
2
stands for the multiplicative inverse of
2x  1.
7?
Can you find the element
Note that
try out all seven possible values of
1
2
7
x
7
in
7?
2 in
such that
Example #2: Translating Representations of Numbers to a Modular System
Find the equivalent representation(s) of these numbers in
1
;
3
33 ;
7.
2
;
3
5;
i
II. Homework
Section 1.4 Homework
PRACTICE PROBLEMS:
1.
Perform the following operations in
5  10;
2.
11 :
5  8;
3 10
Find the equivalent representations of the following numbers in
7
73 ;
To answer this
7.
Thus
2x  1?
1
2
is an
Is there
has only seven elements. If all else fails, you can always just
x.
3
7.
1
;
4
4
;
7
11 .
7;
3
S. I. M.
NAME:_________________________
Mr. Plassmann
DATE:__________________________
BLOCK:________________________
Section 1.4 Homework: Modular Arithmetic
1.
List the elements of
5

2.
5
in a ring. Use the tables below to create an addition and multiplication table for
.
0
1
3
2
4

0
0
1
1
2
2
3
3
4
4
0
1
Use the tables you created above to find the equivalent representations of the following numbers in
2
23 ;
1
;
2
3
;
2
2;
i
3
2
5
.
4
S. I. M.
Mr. Plassmann
Modular Arithmetic
1.
List the elements of
2.
Use the table below to make an addition table for the elements of

0
1
2
3
4
5
6
7
8
9
10
0
11
1
in a ring.
2
3
4
5
11 .
6
7
8
9
10
3.
Do you notice any patterns in the addition table you created?
4.
Use the table below to make a multiplication table for the elements in

0
1
2
3
5
4
11 .
6
7
0
1
2
3
4
5
6
7
8
9
10
5.
Do you notice any patterns in the multiplication table you created?
6.
Can you use the tables you created above to find the following numbers in
5
53 ;
1
;
5
3
;
8
3;
7?
2
8
9
10