Prime and Composite Numbers

What is a Prime Number?

A prime number is a natural number having only two factors i.e. 1 and the number itself. For example, 2, 3, 5, 7, etc.

What is a Composite Number?

A composite number is a natural number greater than 1 that can be expressed as a product of two smaller natural numbers. For example, 4 = 2×2, 6 = 2×3), etc.

How many Prime Numbers are there?

There are infinitly many prime numbers. If we have n prime numbers, then, we can get a new prime number p_n+1 given by [Tex]p_{n+1} = \left( 2\times 3 \times \dots \times p_{n-1} \times p_n \right)+1  [/Tex]where p_i represents the ith prime number (like 2 is the 1st prime number, 3 is the second, 5 is the third one and so on).

Is 1 a Composite Number or Prime Number?

1 is neither a composite number nor a prime number. 1 cannot be written as a product of any two smaller natural numbers (since there are no natural numbers smaller than one) hence it is not composite. 1 does not have two distinct divisors hence it is not a prime number either.

Which is the Smallest Prime and Composite Numbers?

The Smallest Prime and Composite Number is 2 and 4 resepevctively

How to find if a given Natural Number is Prime or not?

Let the number be n. If n = 1, it is not prime. If n ≠1, if there is no factor of n greater than 1 and smaller than or equal to √n then n is a prime number else it is not a prime number.

What is Primality?

The property of a natural number of being prime is known as “primality”.

What are Coprime Numbers?

A pair of natural numbers whose HCF (highest common factor) is 1 are known as coprime numbers. For example, 2 & 3, 1 &2, 4 & 15, etc. A pair of prime numbers is always coprime. Also, a pair of consecutive numbers is always coprime like 5 & 6, 7 & 8, etc.

What are Twin Prime Numbers?

Twin primes are a pair of prime numbers that have a difference of 2 between them. For example, 3 & 5, 5 & 7, 11 & 13, etc.

What is a Primality Test?

A procedure to determine whether a given number is prime or not is known as a primality test.

How to determine whether a given Number is composite?

To determine whether a given number is composite, follow the simple rule: If a given natural number is neither 1 nor prime, then it is a composite number. If the number is not 1, then use a primality test to check whether its prime of not.

Prime and Composite Numbers are commonly used classifications of Natural Numbers based on divisibility and the number of Factors. A Prime Number has only two factors while Composite Numbers have more than two factors. This classification of Numbers makes the study of natural numbers more organized and convenient and is useful in a variety of situations like computer algorithms, biology, understanding of Number Theory, etc.

This article describes what are prime and composite numbers, the types of primes and composite numbers, and tests to check whether a given number is prime or not (primality tests). Finally, a few solved questions, and a few practice problems related to prime and composite numbers are presented.