Number Theory Definition Formulas Properties and Solved Examples for Exams
Number theory, mathematics branch concerned with the properties of positive integers (1, 2, 3, ...). Sometimes referred to as "higher arithmetic," it is one of the oldest and most natural mathematical pursuits. Number theory helps to study the relationship between different types of numbers.
There were no direct applications to the real world of number theory in the past. The emergence of digital computers and digital media showed that number theory could provide surprising answers to real-world problems. Simultaneously, computer technology improvements have made it possible for number theorists to make remarkable advances in factoring large numbers, determining primes, etc.
What is Number Theory
Numbers are theoretically classified into different types, such as natural numbers, whole numbers, complex numbers, and so on. The subclassifications of the natural number are given below:
Odd Numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25. . .
Even Numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 . . .
Square Numbers: 4, 9, 16, 25, 36, 49, 64, 81,100 . . .
Cube Numbers: 8, 27, 64, 125, 216, 343, 512 . . .
Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, . . .
Number Theory Topics
Even Numbers: Integers that can be divided exactly by 2 are called even numbers. Examples of even numbers are 2, 4, 6, 8, 10, 12, 14, 16. These are even numbers as these numbers can easily be divided by 2.
Odd Numbers: Odd numbers are described as any number that is not divisible by 2. Examples of odd numbers 1, 3, 5, 7, 9, 11. 1 is the first positive odd number on the number line.
Square Numbers: The resultant is called a 'Square Number' when a number is multiplied by itself. Square numbers can also be called perfect square numbers. Examples of square numbers are 1, 4, 9, 16, 25, etc.
Cube Numbers: The result of multiplying a whole number by itself twice. Examples of cube numbers are 1, 8, 27, 6, etc. 1, 8, 27, 25, are the cube of 1, 2, 3, 4 respectively.
Prime Numbers: The positive integers with only 2 factors, the number 1 and the integer itself, are prime numbers. For example, factors of 8 are 1,2,4 and 8, which are four factors in total. But factors of 5 are only 1 and 5, totally two. Therefore, 5 is a prime number, but 8 is not, but it is a composite number instead. But always remember that 1 is neither composite nor prime.
Composite Numbers: The numbers that have more than 2 factors are composite numbers, unlike prime numbers that have only 2 factors, i.e. 1 and the number itself. Such numbers are called composite numbers. As they can be divided by more than two numbers, all-natural numbers that are not prime numbers are composite numbers. For instance, 6 is composite, since it is divisible by 1, 2, 3, and 6.
Fibonacci Numbers: The Fibonacci number is a number sequence where a number is the addition of the last two numbers, starting with 0, and 1.
Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
Xn = Xn-1 + Xn-2
Applications of Number Theory
Number theory is used to find out whether a given integer m divides the integer n by some of the significant divisibility tests. Number theory has many applications in both mathematics and practical applications, such as:
Barcodes.
Creation of modular designs.
Method of Memory Management.
Device authentication.
Security system like banking securities.
Websites for E-commerce.
Theory of Coding.
FAQs on Number Theory Concepts Properties and Applications in Mathematics
1. What is Number Theory in mathematics?
Number Theory is the branch of mathematics that studies the properties and relationships of integers (whole numbers). It focuses on topics such as:
- Prime numbers and composite numbers
- Divisibility rules
- Factors and multiples
- Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
- Modular arithmetic and congruences
Number theory is sometimes called the “queen of mathematics” and is widely used in cryptography, coding theory, and computer science.
2. What is a prime number?
A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself. For example:
- 2, 3, 5, 7, 11 are prime numbers.
- 2 is the only even prime number.
A number like 6 is not prime because it has divisors 1, 2, 3, and 6. Prime numbers are the building blocks of number theory.
3. How do you find the Greatest Common Divisor (GCD) of two numbers?
The Greatest Common Divisor (GCD) of two numbers is the largest integer that divides both numbers exactly. You can find it using the Euclidean algorithm:
- Divide the larger number by the smaller.
- Replace the larger number with the smaller number and the smaller number with the remainder.
- Repeat until the remainder is 0.
Example: To find GCD of 48 and 18:
- 48 ÷ 18 = 2 remainder 12
- 18 ÷ 12 = 1 remainder 6
- 12 ÷ 6 = 2 remainder 0
So, GCD(48, 18) = 6.
4. What is the Least Common Multiple (LCM)?
The Least Common Multiple (LCM) of two integers is the smallest positive number that is a multiple of both numbers. One useful formula is:
LCM(a, b) = (a × b) / GCD(a, b)
Example: For 12 and 18:
- GCD(12, 18) = 6
- LCM = (12 × 18) / 6 = 216 / 6 = 36
LCM is commonly used in fraction addition and problem solving involving multiples.
5. What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written uniquely as a product of prime numbers, up to the order of the factors. For example:
- 60 = 2 × 2 × 3 × 5
- 60 = 2² × 3 × 5
This prime factorization is unique, which makes prime numbers essential in number theory.
6. What is modular arithmetic?
Modular arithmetic is a system of arithmetic for integers where numbers “wrap around” after reaching a certain value called the modulus. It is written using congruence notation:
a ≡ b (mod n)
Example:
- 17 ÷ 5 leaves remainder 2
- So, 17 ≡ 2 (mod 5)
Modular arithmetic is widely used in cryptography, clocks (mod 12), and computer algorithms.
7. What is the difference between prime and composite numbers?
The difference between prime numbers and composite numbers lies in the number of their divisors.
- A prime number has exactly two positive divisors: 1 and itself (e.g., 7).
- A composite number has more than two positive divisors (e.g., 12 has 1, 2, 3, 4, 6, 12).
The number 1 is neither prime nor composite because it has only one positive divisor.
8. What are coprime numbers?
Coprime numbers (or relatively prime numbers) are two integers whose GCD is 1. This means they share no common prime factors.
- Example: 8 and 15
- Prime factors of 8: 2 × 2 × 2
- Prime factors of 15: 3 × 5
Since they have no common prime factor, GCD(8, 15) = 1, so they are coprime.
9. What is Euler’s Totient Function?
Euler’s Totient Function, denoted φ(n), counts the number of positive integers less than or equal to n that are coprime to n. If p is prime, then:
φ(p) = p − 1
Example:
- For n = 9, numbers ≤ 9 are 1,2,3,4,5,6,7,8,9
- Coprime with 9: 1,2,4,5,7,8
So, φ(9) = 6. This function is important in modular arithmetic and RSA cryptography.
10. What is a Diophantine equation?
A Diophantine equation is an equation where only integer solutions are required. A common example is a linear Diophantine equation:
ax + by = c
Example:
- 2x + 3y = 7
- One solution is x = 2, y = 1
A linear Diophantine equation has integer solutions if and only if GCD(a, b) divides c.
