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Chapter I

The Problem

Introduction

Modular arithmetic is the arithmetic of congruence, sometimes known informally as "clock

arithmetic". In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity, which

is known as the modulus (which would be 12 in the case of hours on a clock, or 60 in the case of minutes

or seconds on a clock). There are certain generalizations that you can make about the arithmetic of

numbers based on which of these two classes they come from. For example, you know that the sum of

two even numbers is even. The sum of an even number and an odd number is odd. The sum of two odd

numbers is even. The product of two even numbers is even, etc.

Formally, modular arithmetic is the arithmetic of any nontrivial homomorphic image of the ring

of integers. For any such homomorphic image R of Z, there is an integer n such that R is isomorphic tothe ring Z_n of integers modulo n. The addition in the ring Z_n is determined from addition in Z by

computing the remainder, upon division by n, of the sum a+b of two integers a and b. Similarly, for

multiplication in the ring Z_n, one multiplies two integers a and b, and computes the remainder upon

division of ab by n.

For each positive integer n, the ring Z_n has n elements, namely the equivalence relation R that

is defined according to the rule aRb if n divides b-a. It is natural and common to denote the equivalence

class [a] (under the equivalence relation R) of a non-negative integer a

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Statement of the Problem

1. What is the demographic profile of the respondent in terms of?1.1Age1.2Gender1.3Civil Status

2. Do you enjoy solving math problems?3. Who is better when it comes to answering math problems? Male or Female?4. Do you apply the math situation in your life?

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Hypothesis

When we divide two integers we will have an equation that looks like the following:

A / B = Q with remainder R

A is the dividend

B is the divisor

Q is the quotient

Ris the remainder

Sometimes, we are only interested in what the remainderwill be when we divide A by B.

For these cases there is an operator called the modulo operator (abbreviated as mod).

Using the same A, B, Q, and R as above, we would have: A mod B = R

We would say this as "A modulo B is congruent to R". Where B is referred to as the modulus.For example we know: 13/5 = 2 remainder 3

or

13 mod 5 = 3

Visualize modulus with clocks

Observe what happens when we increment numbers by one and then divide them by 3.

0/3=0 remainder 0

1/3=0 remainder 1

2/3=0 remainder 2

3/3=1 remainder 0

4/3=1 remainder 1

5/3=1 remainder 2

6/3=2 remainder 0

...

The remainders start at 0 and increases by 1 each time, until the number reaches one less than the

number we are dividing by.

After that, the sequence repeats.

By noticing this, we can visualize the modulo operator by using circles.

We write 0 at the top of a circle and continuing clockwise writing integers 1,2 ... up to to one less than

the modulus around the circle.

For example, a clock with the 12 replaced by a 0 would be the circle for a modulusof 12.

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To find the result of A mod B = ? we can follow these steps:

1. Construct this clock for size B

2. Start at 0 and move around the clock A steps

3. Wherever we land is our solution.

(if the number is positive we step clockwise, if it's negativewe step counter-clockwise)

Examples

8 mod 4 = ?

With a modulus of 4 we make a clock with numbers 0,1,2,3

We start at 0 and go through 8 numbers in a clockwise sequence 1,2,3,0,1,2,3,0

We ended up at0

so:

8 mod 4 =0

7 mod 2 = ?

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With a modulus of 2 we make a clock with numbers 0,1

We start at 0 and go through 7 numbers in a clockwise sequence 1,0,1,0,1,0,1

We ended up at 1

so:

7 mod 2 = 1

-5 mod 3 = ?

With a modulus of 3 we we make a clock with numbers 0,1,2

We start at 0 and go through 5 numbers in counter-clockwise sequence (5 is negative) 2,1,0,2,1

We ended up at 1

so:

-5 mod 3 = 1

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Scope and Delimitation of Study

The study focuses on the mathematical equations about modular arithmetic. The study also

includes formulas to come up with the solutions and appropriate answer.

The application can be used by anyone in need of understanding the formulas, problems and

solutions about modular arithmetic studies. Most probably, modular arithmetic can be use by future

college student for their future references.

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Definition of Terms

Arithmetic - The branch of mathematics dealing with the properties and manipulation of numbers.

Congruence - congruity: the quality of agreeing; being suitable and appropriate.

Denote - Be a sign of; indicate: "this mark denotes quality". Stand as a name or symbol for.

Divisible - (of a number) capable of being divided by another number without a remainder.

Homomorphic - similarity of form

Modulo - (in number theory) with respect to or using a modulus of a specified number.

Nontrivial - Nontrivial is used formally as an antonym for trivial, in contexts where that word has a

formal meaning.

Notation - A series or system of written symbols used to represent numbers, amounts, or elements in

something such as music or mathematics.

Quantity - The amount or number of a thing not usually estimated by spatial measurement.Remainder - A part, number, or quantity that is left over.

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Chapter II

Review of Literature

Arithmetic for college students

According toKarl J Smith year 2000, (Modular arithmetic is a special type of arithmetic that involves

onlyintegers.This goal of this article is to explain the basics of modular arithmetic while presenting a

progression of more difficult and more interesting problems that are easily solved using modular

arithmetic.

Motivation

Let's use a clock as an example, except let's replace the at the top of the clock with a. Starting at noon,

the hour hand points in order to the following:

This is the way in which we count in modulo 12. When we add to , we arrive back at . The same is

true in any other modulus (modular arithmetic system). In modulo, we count

We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers

from to , when written in modulo 5, are

where is the same as in modulo 5. Because all integers can be expressed as , , , , or in

modulo 5, we give these integers their own name: theresidue classes modulo 5. In general, for a natural

number that is greater than 1, the modulo residues are the integers that arewhole numbers less

than :

This just relates each integer to itsremainder from theDivision Theorem.While this may not seem all

that useful at first, counting in this way can help us solve an enormous array ofnumber theory problems

much more easily!

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8/13/2019 Discrete Topic

9/28

9

Congruence

There is a mathematical way of saying that all of the integers are the same as one of the modulo 5

residues. For instance, we say that 7 and 2 arecongruentmodulo 5. We write this using the symbol :

In other words, this means in base 5, these integers have the same last digit:

2(base 5) 12(base 5) 22(base 5) 32(base 5) 42(base 5)

The (mod 5) part just tells us that we are working with the integers modulo 5. In modulo 5, two integers

are congruent when their difference is amultiple of 5. Thus each of the following integers is congruent

modulo 5:

In general, two integers and are congruent modulo when is a multiple of . In other

words, when is an integer. Otherwise, , which means that

and are not congruent modulo .

Principles of Real Analysis

By S.C. Malik (Author)

One of the benefits of modular congruence is that it behaves an awful lot like the regular "equals"

you're used to playing with. In fact, modular congruence is an equivalence relation, which means it has

the following properties

Reflexive: for any integer aand any modulus m, we have aamodm.

Symmetric: for any integers aand band any modulus m, if abmodmthen bamodm.

Transitive: for any integers a,band c, and any modulus m, if abmodmand bcmodm, then acmodm.

Proof: To prove the reflexive property, note that aamodmjust means that we want to verify maa=0.

We saw a while back, though, that any integer mdivides 0, so this statement is valid.

To prove symmetry, we need to show that abmodmimplies bamodm. If abmodm, though, the

definition of modular congruence tells us that mab, so that mk=ab. But then we

have m(k)=(ab)=ba, and so mba. By the definition of modular congruence, we therefore

have bamodm.

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