This document discusses key concepts in number theory including divisibility, greatest common divisors, least common multiples, prime and composite numbers, relative primality, modular arithmetic, factorials, and applications. It defines these terms and provides examples. Greatest common divisors are the largest integers that divide two numbers. Least common multiples are the smallest integers divisible by two numbers. Prime numbers have only two factors and composite numbers are multiples of primes. Relative primality means two numbers have no common prime factors. Modular arithmetic uses the remainder of a division. Factorials are the product of integers up to a given number. Applications include cryptography.

1. Md.Arifur Rahman
Id:163015004
Number Theory

2. Content
1) Divisibility
2) Greatest common divisor
3) Least Common Multiples
4)Prime and composite number
5) Relative prime
6) MOD:Congruence relation
7) Factorial number
8) Applications of Number Theory

3. Divisibility

4. Greatest Common Divisors
Let a and b be integers, not both zero.
The largest integer d such that d | a and d | b is called the greatest
common divisor of a and b.
The greatest common divisor of a and b is denoted by gcd(a , b).
Example :
GCD (12, 18) =6

5. Least Common Multiples
• The least common multiple of the positive integers a
and b is the smallest positive integer that is divisible by
both a and b.
• We denote the least common multiple of a and b by
lcm(a, b).
•Example:
lcm(12, 18) =36

6. Primes and composite
•A positive integer p is called prime if it has just two
divisor,namely 1 and p..
•A composite number is a multiple of prime number.
Example: prime: 2=2*1
2=1*2
Composite: 15 = 3*5
/
prime prime

7. Relative prime
• When gcd(m , n)=1 ,the integers m and n have no
prime factors in common and we say that they’re
relative prime.
• Example: are 10, 17, and 21 pairwise relatively prime?
– gcd(10,17) = 1, gcd (17, 21) = 1, and gcd (21, 10) = 1
– Thus, they are pairwise relatively prime
– Since gcd(10,24) ≠ 1, they are not
7

8. Modulo Operation
rnqaqrna  s.t.,mod

where 0  r  n 1
Example:
7 mod 3 = 1

9. Written n!, the product of all positive
integers less than and equal to n.
Ex: Evaluate.
   ! 1 2 ... 1n n n n      
8!
5!  3368 7 6  8 7 6 5 4 3 2 1
5 4 3 2 1
      
    
Factorial

10. The zero
factorial is one
0! = 1
Zero Factorial

11. 11
Real life Applications
1. Private and public key cryptography
2. Encrypting a message
3. Decrypting a message
4. Cracking a message
5. Quantum computers

12. www.google.com
www.wikipedia.com
www.slideshare.net
References

13. Thanks For Being
With me.