This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.

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2. Submitted by:Submitted by:
TEENA ZACHARIASTEENA ZACHARIAS
class:1BEdclass:1BEd
Option: MathematicsOption: Mathematics
Roll No:29Roll No:29
Mangalam College of EducationMangalam College of Education
Submitted to:Submitted to:
Mrs.Sijji JoseMrs.Sijji Jose
Dept. Educational technologyDept. Educational technology

3. Introduction toIntroduction to NumberNumber
TheoryTheory
Number theory is aboutNumber theory is about
integersintegers and their properties.and their properties.
We will start with the basicWe will start with the basic
principles ofprinciples of
1.1.Divisibility,Divisibility,
2.2.Greatest common divisors,Greatest common divisors,
3.3.Least common multiples,Least common multiples,
4.4.Modular arithmeticModular arithmetic..
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4. DivisionDivision
If a and b are integers with aIf a and b are integers with a ≠≠ 0, we say0, we say
that athat a dividesdivides b if there is an integer c sob if there is an integer c so
that b = ac.that b = ac.
When a divides b we say that a is aWhen a divides b we say that a is a factorfactor ofof
b and that b is ab and that b is a multiplemultiple of a.of a.
The notationThe notation a | ba | b means that a divides b.means that a divides b.
We writeWe write a X ba X b when a does not divide b.when a does not divide b.
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5. Divisibility TheoremsDivisibility Theorems
For integers a, b, and c it is true thatFor integers a, b, and c it is true that
 If a | b and a | c, then a | (b + c)If a | b and a | c, then a | (b + c)
Example:Example: 3 | 6 and 3 | 9, so 3 | 15.3 | 6 and 3 | 9, so 3 | 15.
 If a | b, then a | bc for all integers cIf a | b, then a | bc for all integers c
Example:Example: 5 | 10, so 5 | 20, 5 | 30, 5 | 40, …5 | 10, so 5 | 20, 5 | 30, 5 | 40, …
 If a | b and b | c, then a | cIf a | b and b | c, then a | c
Example:Example: 4 | 8 and 8 | 24, so 4 | 24.4 | 8 and 8 | 24, so 4 | 24.
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6. The Division AlgorithmThe Division Algorithm
LetLet aa be an integer andbe an integer and dd a positive integer.a positive integer.
Then there are unique integersThen there are unique integers qq andand rr, with, with
00 ≤≤ r < dr < d, such that, such that a=dq+ra=dq+r..
In the above equation,In the above equation,
• dd is called the divisor,is called the divisor,
• aa is called the dividend,is called the dividend,
• qq is called the quotient, andis called the quotient, and
• rr is called the remainderis called the remainder..
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7. The Division AlgorithmThe Division Algorithm
Example:Example:
When we divide 17 by 5, we haveWhen we divide 17 by 5, we have
17 = 517 = 5⋅⋅3 + 2.3 + 2.
• 17 is the dividend,17 is the dividend,
• 5 is the divisor,5 is the divisor,
• 3 is called the quotient, and3 is called the quotient, and
• 2 is called the remainder.2 is called the remainder.
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8. The Division AlgorithmThe Division Algorithm
Another example:Another example:
What happens when we divide -11 by 3 ?What happens when we divide -11 by 3 ?
Note that the remainder cannot be negative.Note that the remainder cannot be negative.
-11 = 3-11 = 3⋅⋅(-4) + 1.(-4) + 1.
• -11 is the dividend,-11 is the dividend,
• 3 is the divisor,3 is the divisor,
• -4 is called the quotient, and-4 is called the quotient, and
• 1 is called the remainder.1 is called the remainder.
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9. Greatest Common DivisorsGreatest Common Divisors
Let a and b be integers, not both zero.Let a and b be integers, not both zero.
The largest integer d such that d | a and d | b isThe largest integer d such that d | a and d | b is
called thecalled the greatest common divisorgreatest common divisor of a and b.of a and b.
The greatest common divisor of a and b is denotedThe greatest common divisor of a and b is denoted
by gcd(a, b).by gcd(a, b).
Example 1:Example 1: What is gcd(48, 72) ?What is gcd(48, 72) ?
The positive common divisors of 48 and 72 areThe positive common divisors of 48 and 72 are
1, 2, 3, 4, 6, 8, 12, 16, and 24, so gcd(48, 72) = 24.1, 2, 3, 4, 6, 8, 12, 16, and 24, so gcd(48, 72) = 24.
Example 2:Example 2: What is gcd(19, 72) ?What is gcd(19, 72) ?
The only positive common divisor of 19 and 72 isThe only positive common divisor of 19 and 72 is
1, so gcd(19, 72) = 1.1, so gcd(19, 72) = 1.
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10. Least Common MultiplesLeast Common Multiples
Definition:Definition:
TheThe least common multipleleast common multiple of the positiveof the positive
integers a and b is the smallest positiveintegers a and b is the smallest positive
integer that is divisible by both a and b.integer that is divisible by both a and b.
We denote the least common multiple of aWe denote the least common multiple of a
and b by lcm(a, b).and b by lcm(a, b).
Examples:Examples:
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lcm(3, 7) =lcm(3, 7) = 2121
lcm(4, 6) =lcm(4, 6) = 1212
lcm(5, 10) =lcm(5, 10) = 1010

11. GCD and LCMGCD and LCM
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a = 60 =a = 60 = 2222
3311
5511
b = 54 =b = 54 = 2211
3333
5500
lcm(a, b) =lcm(a, b) = 2222
3333
5511
= 540= 540
gcd(a, b) =gcd(a, b) = 2211
3311
5500
= 6= 6
Theorem: ab =Theorem: ab = gcd(a,b)lcm(a,b)gcd(a,b)lcm(a,b)

12. Modular ArithmeticModular Arithmetic
Let a be an integer and m be aLet a be an integer and m be a
positive integer.positive integer.
We denote byWe denote by a mod ma mod m thethe
remainder when a is divided by m.remainder when a is divided by m.
Examples:Examples:
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9 mod 4 =9 mod 4 = 11
9 mod 3 =9 mod 3 = 00
9 mod 10 =9 mod 10 = 99
-13 mod 4 =-13 mod 4 = 33

13. THANKS…THANKS…
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