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# Prime and Composite Numbers

Mathematics is known as the mother of all other sciences because mathematics provides tools to solve problems of other branches of science. While studying mathematics, we have to deal with numbers. These numbers are divided into different categories like rational numbers, irrational numbers, natural numbers, even numbers, whole numbers, odd numbers, prime numbers, and composite numbers. If we want to learn mathematics effectively, we should learn these numbers. Its reason is that these numbers provide us a base to understand other concepts of mathematics. Here, we will study full detail about prime numbers and composite numbers.

## What are Prime Numbers?

Different people define prime numbers in different ways. The most famous definitions of prime numbers are given below;

♦ A number that is divisible by ‘1’ and itself is known as a prime number. This number should be greater than ‘1’. It means ‘1’ is not a prime number.

♦ A whole number that we can’t make by multiplying other whole numbers is known as a prime number.

♦ A number that has only two factors is known as a prime number.

### Examples of Prime Numbers

Some essential examples of prime numbers are 2, 3, and 7, etc. These three examples satisfy all three definitions. It means 2, 3, and 7 are divisible by ‘1’, and itself only, we can’t make 2, 3, and 7 by multiplying two whole numbers and 2, 3 and 7 have two factors only.

## What are Composite Numbers?

Like prime numbers, there are also different definitions of composite numbers. Here, we have provided the most famous definitions of composite numbers.

♦ The numbers which are divisible by ‘1’, by itself, and by another number are known as composite numbers. The composite numbers are also greater than ‘1’. It means ‘1’ is not a composite number.

♦ A whole number that is made by multiplying another whole number is known as a composite number.

♦ A number that has more than two factors is known as a composite number. These factors are ‘1’, the number itself, and some other numbers.

### Examples of Composite Numbers

4, 6, and 9 are three examples of composite numbers. These three examples of composite numbers also satisfy the above three definitions of composite numbers. Its reason is that 4, 6, and 9 are divisible by ‘1’, by itself and by another number, we can make 4, 6, and 9 by multiplying two whole numbers, and 4, 6, and 9 have more than two factors.

### Expert Tip

Most of the students face lots of problems to differentiate between prime numbers and composite numbers. According to experts, prime numbers and composite numbers are opposite to each other. You just need to learn one of them. For example, if you know the concept of prime numbers, you can also find other numbers. Its reason is that after finding the prime numbers from the given set of numbers, the remaining numbers will be composite numbers. Similarly, if you know the basic concept of composite numbers, you can also find the prime numbers.

## Is ‘1’ a Composite Number or a Prime Number?

In the definitions, we have provided a clear answer to this question but some students have confusion about ‘1’ because they can’t decide whether it is a prime number or it is a composite number. For example, if we ask the students to write prime numbers between 1 to 10, in the list of prime numbers, they also write ‘1’. Its reason is that they consider that it doesn’t have more than two factors. They should know that ‘1’ is neither a prime number nor a composite number. Therefore, you should not write it in the list of prime numbers or in the list of composite numbers.

## How to Determine if a Number is a Prime Number or a Composite Number?

After understanding the basic definitions of prime numbers and composite numbers, the next step is to determine whether a number is a prime number or a composite number. There are various ways to do this. We have explained the most important ways.

### · Using Factorization

By finding the prime factors of a number, you can easily know whether a number is a prime number or a composite number. We try to understand it with the help of examples. Let us consider we have a number 27 and we have to find whether it is a prime number or a composite number. First, we find the factors of this number. The factors of this number are 1, 3, 9, and 27. It means this number has more than two factors. Therefore, it is a composite number. Similarly, if we have a number 13, find either it is a prime number or a composite number. The factors of this number are 1 and 13. As this number has only two factors, therefore, it is a prime number. By following the same technique, you can easily differentiate between prime and composite numbers.

### ·  Using the Calculator

You can also find whether a number is a prime number or a composite number by using a calculator. If the given number is an even number, it is a composite number because an even number is divisible by 2. If a number is divisible by ‘1’, itself and ‘2’, it means it has more than two factors. Therefore, it is a composite number. We can also say that all the even numbers are also composite numbers.
On the other hand, if you have an odd number, you will have to use the concept of divisibility to determine whether it is a prime number or a composite number. Let us consider, you have the number ‘39’. When you divide it by 2, you will get the answer 19.5 which is not a whole number. When you divide it by 3, you will get the answer 13 which is a whole number. This divisible test tells us that 39 has more than 2 factors. Therefore, it is also a composite number.

## Key Points

Some key points relevant to these two essential kinds of numbers are given below;

i. We use the English alphabet ‘P’ to represent the set of Prime numbers.

ii. We use the English alphabet ‘C’ to represent the set of Composite numbers.

iii. ‘1’ is not the element of prime and composite sets.

iv. Prime and composite numbers are opposite to each other.

v. All the even numbers are also composite numbers but odd numbers can be prime or composite.

## Solution of the Questions Relevant to the Prime Numbers and Composite Numbers

After reading extensive discussions about these two essential types of numbers, you can easily find out prime and composite numbers from a given set of numbers. For this reason, we try to solve some problems.
1. Find prime numbers and composite numbers between 55 and 75.
Prime Numbers = 59, 61, 67, 71, 73
Composite Numbers = 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74
2. Find prime and composite numbers between 16 to 24.
Prime Numbers = 17, 19, 23
Composite Numbers = 18, 20, 21, 22
3. Find 3 consecutive prime numbers just below 33?
Prime Numbers = 31, 29, and 23.
4. Find 4 consecutive composite numbers just below 60?
Composite Numbers = 58, 57, 56, and 55.