This document introduces the concepts of number theory, including divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines divisibility as an integer a dividing another integer b if b can be written as a product of a and another integer. The greatest common divisor of two integers is the largest integer that divides both, while the least common multiple is the smallest positive integer divisible by both. Modular arithmetic involves finding the remainder of dividing an integer by a positive integer. Examples are provided to illustrate these key number theory topics.

1. 1

3. Introduction toIntroduction to NumberNumber TheoryTheory
Number theory is aboutNumber theory is about integersintegers and theirand their
properties.properties.
We will start with the basic principles ofWe will start with the basic principles of
1.1.Divisibility,Divisibility,
2.2.Greatest common divisors,Greatest common divisors,
3.3.Least common multiples,Least common multiples,
4.4.Modular arithmetic.Modular arithmetic.
3

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
of b and that b is aof b 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.
4

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 | 63 | 6 andand 3 | 93 | 9, so, so 3 | 153 | 15..
 If a | b, then a | bc for all integers cIf a | b, then a | bc for all integers c
Example:Example: 5 | 105 | 10, so, so 5 | 205 | 20,, 5 | 305 | 30,, 5 | 405 | 40, …, …
 If a | b and b | c, then a | cIf a | b and b | c, then a | c
Example:Example: 4 | 84 | 8 andand 8 | 248 | 24,, soso 4 | 244 | 24..
5

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 remainder.is called the remainder.
6

7. The Division AlgorithmThe Division Algorithm
Example:Example:
When we divideWhen we divide 1717 byby 55, we have, we have
17 = 517 = 5⋅⋅3 + 2.3 + 2.
• 1717 is the dividend,is the dividend,
• 55 is the divisor,is the divisor,
• 33 is called the quotient, andis called the quotient, and
• 22 is called the remainder.is called the remainder.
7

8. The Division AlgorithmThe Division Algorithm
Another example:Another example:
What happens when we divideWhat happens when we divide -11-11 byby 33 ??
Note that the remainder cannot be negative.Note that the remainder cannot be negative.
-11 = 3-11 = 3⋅⋅(-4) + 1.(-4) + 1.
• -11-11 is the dividend,is the dividend,
• 33 is the divisor,is the divisor,
• -4-4 is called the quotient, andis called the quotient, and
• 11 is called the remainder.is called the remainder.
8

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.
9

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 positive integerintegers a and b is the smallest positive integer
that is divisible by both a and b.that is divisible by both a and b.
We denote the least common multiple of a and bWe denote the least common multiple of a and b
by lcm(a, b).by lcm(a, b).
Examples:Examples:
10
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
11
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 a positive integer.Let a be an integer and m be a positive integer.
We denote byWe denote by a mod ma mod m the remainder when a isthe remainder when a is
divided by m.divided by m.
Examples:Examples:
12
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…
13