What are Prime and Composite Numbers?
Prime Numbers
A prime number is a natural number greater than 1 that has exactly two distinct factors:
- 1
- the number itself
Examples:
2, 3, 5, 7, 11, 13, 17…
Example explanation:
7 has factors → 1 and 7 only → Prime number
Composite Numbers
A composite number is a natural number greater than 1 that has more than two factors.
Examples:
4, 6, 8, 9, 10, 12, 15…
Example explanation:
6 has factors → 1, 2, 3, 6 → Composite number
Important Note About 1
Number 1 is neither prime nor composite because it has only one factor (1).
List of Prime and Composite Numbers (1 to 50)
Prime Numbers (1–50)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Composite Numbers (1–50)
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30…
How to Check if a Number is Prime
Method 1: Factor counting
Example:
Check 11
Factors:
1, 11 → only two → Prime
Method 2: Divisibility test
Check divisibility by numbers less than √n.
Example:
Check 29
√29 ≈ 5.38
Check divisibility by:
2, 3, 5
29 not divisible → Prime
Smallest and Largest Prime Numbers
Smallest prime number = 2
There is no largest prime number because primes continue infinitely.
Even Prime Numbers
2 is the only even prime number.
All other even numbers are divisible by 2 → composite.
Examples of even composite numbers:
4, 6, 8, 10…
Properties of Prime Numbers
- Prime numbers are greater than 1
- They have only two factors
- Except 2, all primes are odd
- There are infinitely many prime numbers
- Every composite number can be expressed as product of primes
Properties of Composite Numbers
- Composite numbers have more than two factors
- Every composite number can be expressed as multiplication of primes
- Smallest composite number = 4
Prime Factorisation
It means to break a number into product of prime numbers.
Example:
24 = 2 × 2 × 2 × 3
Prime factorisation:
24 = 2³ × 3
Fundamental Theorem of Arithmetic
Every composite number can be expressed as product of prime numbers in a unique way.
Example:
36 = 2² × 3²
Even if written differently:
36 = 2 × 2 × 3 × 3
Prime factors remain same.
Number of prime factors of given number
Example:
For the number 60, the prime factorisation is:
60 = 2 × 2 × 3 × 5
So, how many prime numbers are there?
There are 4 prime factors (counting repetition):
2, 2, 3, 5 → 4 numbers
But there are 3 different (distinct) prime numbers:
2, 3, 5 → 3 numbers
Co-prime Numbers
Two numbers are co-prime if their HCF is 1.
Examples:
8 and 15
Factors of 8:
1, 2, 4, 8
Factors of 15:
1, 3, 5, 15
Common factor = 1
Twin Prime Numbers
Prime numbers having difference of 2.
Examples:
3 and 5
5 and 7
11 and 13
17 and 19
Sieve of Eratosthenes (Finding Prime Numbers)
Steps:
- Write numbers from 1 to 100
- Remove multiples of 2
- Remove multiples of 3
- Continue process
Remaining numbers are prime.
Relationship Between Prime and Composite Numbers
Every number greater than 1 is either:
- prime
OR - composite
Exception:
1 is neither.
Applications of Prime Numbers
Prime numbers are used in:
- Cryptography (internet security)
- Computer algorithms
- Coding theory
- Mathematics research
- Banking security
- Digital signatures
Example:
RSA encryption uses prime numbers.
Common Mistakes
Mistake 1:
Thinking 1 is prime.
Correction:
1 is neither prime nor composite.
Mistake 2:
Thinking all odd numbers are prime.
Example:
9 is odd but composite.
Quick Trick to Identify Prime Numbers (Small Numbers)
Check divisibility by:
2, 3, 5, 7
If not divisible → likely prime.
Practice Examples
Example 1:
Is 19 prime?
Factors:
1, 19 → Prime
Example 2:
Is 21 prime?
21 = 3 × 7 → Composite
Example 3:
Prime factorisation of 60
60 = 2² × 3 × 5