Composite and Prime Numbers: Numbers are used in our daily life in many instances like doing calculations when buying or selling something, making payment, asking about the time, the number of runs made by the India team during the cricket match, counting the number of students sitting in a classroom, marks obtained by a student, etc.

Numbers play an important role in Mathematics as they act as the building blocks. The numbers are classified into different types, such as odd numbers, even numbers, composite numbers, and prime numbers. This article deals with prime and composite numbers, properties, examples, and applications in real life.

Definition of Composite Numbers

In basic Mathematics, the numbers that have more than two factors are known as composite numbers.
For example, factors of \({\rm{6}}\) are \({\rm{1,}}\,{\rm{2,}}\,{\rm{3}}\) and \({\rm{6}}\). Thus, there are four factors in total, and \(6\) is a composite number.

The first ten composite numbers are \(4,\,6,\,8,\,9,\,10,\,12,\,14,\,15,\,16,\,18\). To understand better, have a look at the following examples.

Definition of Prime Numbers

The numbers that have only two factors, \(1\), and the number itself are known as prime numbers.
For example, factors of \(7\) are only \(1\) and \(7\), two in total. Hence, \(7\) is a prime number.

The first ten prime numbers are \(2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,29\). To better understand, have a look at the following example.

Properties of Composite and Prime Numbers

Knowing some basic properties are essential to identify a prime number or a composite number.

Properties of Composite Numbers

Some properties of composite numbers are:

1. Composite numbers have more than two factors.
2. We can get a composite number if we multiply two or more two positive integers other than \(1\).
3. A composite number is divisible by a composite number as well as a prime number.
4. A composite number can be formed by the product of two or more prime numbers.

Properties of Prime Numbers

Some of the properties of prime numbers are:

1. A prime number can have only two factors.
2. All prime numbers are odd except \(2\) or we can say that \(2\) is the only even prime and the smallest prime number.
3. Two prime numbers are always coprime to each other. It means the HCF of two prime numbers is always \(1\).

Composite Numbers List from \(1\) to \(100\) Chart

There are many composite numbers in the number system. Here is the list of composite numbers from \(1\) to \(100\). In the following chart, the boxes coloured in blue are the composite numbers.

Prime Numbers List from \(1\) to \(100\) Chart

There are many prime numbers in the number system. In the following chart, the boxes coloured in blue are the prime numbers present between \(1\) and \(100\).

Difference Between Prime and Composite Numbers

There are some differences between the prime numbers and the composite numbers.

Prime Numbers	Composite Numbers
It is only divisible by\(1\) and itself. Therefore, it has only two factors.	It is divisible by more than two numbers. Therefore, it has more than two factors.
It can only be written as a product of two numbers ( \(1\) and the number itself )	It can be written as the product of two or more numbers
There is only one even prime number.	There are many even composite numbers.

Prime Factorisation of Composite and Prime Numbers

Prime factorisation is a procedure to get the prime factors of a given number. Prime factors are factors of a number that are prime numbers.

Let us apply prime factorization for some composite and prime numbers.

Composite Numbers	Prime Factorisation
\(4\)	\(2\,\, \times \,\,2\)
\(6\)	\(2\,\, \times \,\,3\)
\(8\)	\(2\,\, \times \,\,2\,\, \times \,2\)
\(9\)	\(3\,\, \times \,\,3\)
\(10\)	\(2\,\, \times \,\,5\)
\(12\)	\(2\,\, \times \,\,2\,\, \times \,3\)
\(14\)	\(2\,\, \times \,\,7\)
\(15\)	\(3\,\, \times \,\,5\)
\(16\)	\(2\,\, \times \,\,2\,\, \times \,\,2\,\, \times \,\,2\)
\(18\)	\(2\,\, \times \,\,3\,\, \times \,3\)
\(20\)	\(2\,\, \times \,\,2\,\, \times \,5\)

Prime Numbers	Prime Factorisation
\(2\)	\(1\,\, \times \,\,2\)
\(3\)	\(1\,\, \times \,\,3\)
\(5\)	\(1\,\, \times \,\,5\)
\(7\)	\(1\,\, \times \,\,7\)
\(11\)	\(1\,\, \times \,\,11\)
\(13\)	\(1\,\, \times \,\,13\)
\(17\)	\(1\,\, \times \,\,17\)
\(19\)	\(1\,\, \times \,\,19\)

Uses of Prime and Composite Numbers in Real life

There are some real-life applications of the composite and prime numbers.

For example, whenever someone purchases any item online that involves a credit card, prime and composite numbers participate in the transaction process.  The computer uses a code to identify a person’s credit card.

Before the buyer’s information of the credit card is sent through cyberspace, it is encrypted to protect it from being hacked by someone. To make sense of the encrypted information, a key to the code is needed. One of the most popular methods of encryption makes use of prime and composite numbers.

We know that a prime number is only divisible by itself and the number one, and we get a composite number when we multiply two prime numbers together. This concept is used for encryption codes by multiplying two prime numbers together.

The composite number is identified by the computer, but only the bank knows the two original prime numbers. The composite numbers are used as codes that are usually extremely large. Because there is an unknown number of prime numbers, it is impossible to break down the composite number into its two prime factors. Without knowing the two exact prime numbers, a hacker cannot get the credit card information.

Solved Example Questions on Composite & Prime Numbers

Q.1. Write the first ten composite numbers.
Ans: First ten composite numbers are \(4,\,6,\,8,\,9,\,10,\,12,\,14,\,15,\,16,\,18,\,20.\)

Q.2. Identify the prime numbers from the following \(23,\,12,\,60,\,51.\)
Ans: Given numbers are \(23,\,12,\,60,\,51.\)
Let us check which are prime numbers by finding the prime factors of the numbers.
\(23\,\, = \,\,1 \times 23, 12\, = \,2\,\, \times \,\,2\,\, \times \,3, 60\,\, = \,2\,\, \times \,\,2\, \times \,3\, \times 551 = 3\,\, \times \,\,17\)
Hence,\(23\) is the prime number among the given numbers as it has two factors such as \(1\) and the number itself.

Q.3. Find the prime numbers between \(20\) and \(40.\)
Ans: The prime numbers between \(20\) and \(40\) are \(23,\,29,\,31,\,37.\)

Q.4. If \(X\) is a prime number, then find its factor.
Ans: if \(X\) is a prime number, then its factor is \(1,X\) as \(1 \times X = X.\)

Q.5. Is \(2\) a composite number?
Ans: No,\(2\) is not a composite number as \(2\) has two factors, such as \(1\) and the other is \(2\) itself.

Summary

In this article, we have learned about composite numbers and prime numbers and their properties. In addition, we have discussed their examples and the method to identify composite numbers and prime numbers.

FAQs Composite and Prime Numbers

Q.1. Write few examples of composite numbers and prime numbers.
Ans: The examples of composite numbers are \(4, 6, 8, 10, 12, 14, 15\) etc. as they have more than two factors.
Examples of prime numbers are \(2, 3, 5, 7, 11, 13\) etc. as they have only two prime factors.

Q.2. Is \(0\) a prime or composite number?
Ans: No,\(0\) is neither prime nor composite.We know that \(0\) multiplied by any number results in \(0\). This property is called the property of zero (\(0\)) for multiplication. This property or analogy cannot be applied in the case of factorisation.

Q.3. \(9\) Is a composite number?
Ans: \(9\) is a composite number because it has more than two factors, such as\(1,\,3\)  and or we can say \(9\) can be divided by three numbers such as \(1,\,3\) and \(9\).

Q.4. Why \(4\) is the smallest composite number?
Ans: is not a composite number as the divisor of \(1\) is \(1\). The positive integers \(2\) and \(3\) are prime numbers because they are divisible by only two factors,\(1\) and itself. Thus, \(2\) and \(3\) are not composite numbers. However, for the number\(4\), we get three factors in total. Therefore, the divisors of \(4\) are \(1,\,\,2,\,\,4\). So, this number fulfils the condition of a composite number as discussed above.

Q.5. How to find the prime numbers?
Ans: The following method is used to find the prime numbers.
It is known that \(2\) is the only even prime number. \(1\) is not a prime number. Only two consecutive natural numbers are prime numbers these are \(2\) and \(3\). Excluding these two numbers, every prime number can be formed as \(\left({6n + 1} \right)\) or \(\left({6n – 1} \right)\) (except the multiples of prime numbers, that is \(2,\,3,\,5,\,7,\,11\)) where \(n\) is a natural number.
For example: when \(n\, = \,1\) we will \(5,\,7\)
\(6(1)\,\, – \,\,1\,\, = \,\,5, 6(1)\,\, + \,\,1\,\, = \,\,7\)
When \(n\, = \,2\) we will get \(11,\,13.\,6\left( 2 \right) – 1 = 11,\,6\left( 2 \right) + 1 = 13\)
When \(n\, = \,3\) we will get \(17,\,19.\,6\left( 3 \right) – 1 = 17,\,6\left( 3 \right) + 1 = 19\)
When \(n\, = \,4\) we will get \(23,\,\,25\)
\(6(4)\, – 1 = \,23\)
\(6(4)\, + 1 = \,25\) (multiple of \(5\))
Here we can see that \(25\)) is the multiple of \(5\)) and which is not a prime number.
Thus, we can get the prime numbers using the formulas( if we get multiples of \(2,\,3,\,5,\,7,\,11\) we will not consider them).

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