280 LearnersLast updated on 10 September 2025
Prime Numbers
Do you know what prime numbers are? They are the special numbers or integers greater than 1. We can’t break them into a multiplication of two smaller whole numbers. This cannot be written as the product of two distinct positive integers. In this article, let’s explore what prime numbers are.
What are Prime Numbers
A prime number is defined as any positive number that can be divided only by itself and 1. They are one of the most essential building blocks of mathematics. Now, the prime numbers are used in decryption and encryption software, in rotor machines, in telecommunication codes, and in hash tables that organize and display the data. For example, consider the prime number 7. It only has divisors: 1 and 7. 7 ÷ 2 = 3.5, with a remainder of 1. Here, dividing a prime number by any other natural number results in other leftover numbers
Key Concepts Related to Prime Numbers
Prime numbers are the basis of many mathematical functions. In this section, let’s discuss the key concepts related to prime numbers.
Smallest prime number:
The smallest prime is 2, which is also the only even prime number.
Odd prime number:
All prime numbers that are greater than two are called odd numbers.
Composite numbers:
Composite numbers are which is greater than one that are not prime are called
composite numbers. (they have more than two factors).
Number 1: 1 is not a prime number because it has only one factor (itself).
Primality test: To check whether a number n is prime or not, we only need to test divisibility by prime numbers up to n.
Infinite nature:
There are infinitely many prime numbers.
Prime factorization:
Each of the composite numbers can be expressed as the product of two or more prime numbers.
Twin Primes:
A pair of prime numbers that differ by two is called a twin prime.
Distribution:
As numbers increase in size, the frequency of prime numbers decreases. But they never stop occurring.
Prime and Composite Numbers
Numbers can sometimes be either classified as prime or composite based on their factors.
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Prime number |
Composite number |
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Natural numbers are greater than one and have only two factors that are one and the number itself. |
Natural numbers greater than one and have more than two factors. |
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Prime numbers cannot be written as the product of two smaller integers. For example: 5 = 5 × 1 |
The numbers that can be expressed as the product of prime numbers are composite numbers. For example: 12 = 2 × 2 × 3 = 22 × 3 |
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The numbers which are only divisible by 1 and the number itself are the prime numbers |
Composite numbers have more than two factors. Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 |
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Examples: 2, 3, 5, 7, 11, etc. |
Examples: 4, 6, 8, 10, 12, etc. |
Twin Primes
A pair of prime numbers where the difference between them is two is known as twin primes. In between a pair of twin primes, there is a composite number. For example, (3, 5), (5, 7), (11, 13), (17, 19), and so on.
Co-prime Numbers
Co-primes are a pair of numbers where the common factor of the two numbers is 1. In co-primes, one number can be prime, and the other can be a composite number. Here, the only condition is that the common factors should be 1. For example, (2, 3), (5, 9), (9, 10), and so on.
Prime Factorization
Any composite number can be expressed as the product of its prime factors. This process is known as prime factorization. For example, the prime factorization of 34 is 2 × 17; here, 2 and 17 are the prime factors of 34.
Common Mistakes and How to Avoid Them in Prime Numbers
When learning about prime numbers, students make errors that they tend to repeat. To learn and master prime numbers, let’s discuss some common mistakes and how to avoid them.
Mistake 1
Considering All Odd Numbers as Prime Numbers
Students assume that all odd numbers are prime, as they are not divisible by 2. But not all odd numbers are prime, for example, 9, 15, 21, and so on. So to avoid this error, students should check the factors of the numbers.
Mistake 2
Checking the Divisibility Only With Small Primes
Students often consider a large number to be prime without verifying it properly. They only check the divisibility with the smallest prime numbers, such as 2, 3, and 5. To avoid it in the division method, they should check the divisibility with all the prime numbers less than or equal to the square root of the number. For example, to check if 257 is a prime number, students should check for divisibility up to the square root of 257. √257 = 16. 03, that is till 16.
Mistake 3
Thinking that negative numbers are Prime
Sometimes the definition of prime numbers is confusing; that is, all the numbers that have only two factors, one and the number itself. Students may consider -2, -3, -5, -7, and so on, as prime numbers. Prime numbers are positive numbers greater than 1 with only two factors, that is, one and the number itself.
Mistake 4
Misunderstanding the concept of twin primes
When identifying twin primes, students may identify the wrong pairs. For example, considering 15 and 17 as twin primes, the difference here is 2. To avoid the error, students should understand that in twin primes, both numbers should be prime and the difference between them should be 2.
Mistake 5
Misunderstanding the concept of Prime and co-primes.
Confusing co-primes and prime numbers is a common mistake. Students often think that co-primes should be prime numbers, but this is incorrect. Co-primes can be any numbers where the most significant common factor is one.
For example, 8 and 15 are co-primes, as their GCF is 1
Tips and Tricks to Master Prime Numbers
Confused about how to learn and master prime numbers? When learning prime numbers, try to follow these tips and tricks to master them. These tricks can help you learn prime numbers quickly.
- All the even numbers except 2 are non-prime numbers because it has more than two factors. For example, 256 is not a prime number, as it has more than two factors. The common factor between any two prime numbers is 1. So all prime numbers are co-primes.
- The two prime numbers whose difference is 2 are twin primes.
- All composite numbers can be expressed as a product of prime numbers.
- No prime numbers end in 5 except five, as it is divisible by 5.
- The numbers ending in 0 are not prime numbers, as they are divisible by 2 and 10.
- All the even numbers except two can be expressed as the sum of two odd numbers. For example; 10 = 5 + 5; 8 = 5 + 3; 16 = 11 + 5.
Real-world Applications of Prime Numbers
In our daily lives, we use prime numbers in various fields, such as cryptography and data security, as well as in analyzing number patterns and other applications. Let’s learn a few real-world applications of prime numbers.
Cryptography and Data Security: Prime numbers are used in encryption algorithms through public-key cryptography, such as RSA. Cryptography is used to decrypt or encrypt data. For data security, prime numbers can be used as passwords.
Simplification of fractions: Prime numbers can be used to find the GCF of more than two numbers. GCF is used to simplify the fractions.
Analyzing number patterns: As prime numbers are considered the building blocks of all integers, they help to understand the relationship between numbers. We use prime numbers in hash tables to distribute the data key evenly.
Solved Examples of Prime Numbers
Problem 1
What is the sum of the first five prime numbers?
The sum of the first five prime numbers is 28
Explanation
The first five prime numbers are 2, 3, 5, 7, 11
Sum of first five number = 2 + 3 + 5 + 7 + 11 = 28
Problem 2
How many primes are there between 2 and 20
The prime numbers between 2 and 20 are 2, 3, 5, 7, 11, 13, 17, and 19.
Explanation
The prime numbers are the numbers which are only divisible by 1 and the number itself.
So the prime numbers between 2 and 20 are 2, 3, 5, 7, 11, 13, 17, and 19.
Problem 3
If x is a prime number how many factors does x2 have?
If x is a prime number, x2 has 3 factors.
Explanation
Here, x is a prime number
So, the factors of x are 1 and x
The factors of x2 are 1, x, and x2
Therefore, the number of factors of x2 is 3.
Problem 4
Which of the following is not a prime number 83, 101, 105, 89, 11.
105 is not a prime number.
Explanation
The factors of 83, 101, 89, and 11 are 1, and the numbers itself.
Meanwhile, the factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.
Therefore, 105 is not a prime number.
Problem 5
Which is the greatest prime number between 11 and 30?
The greatest prime number between 11 and 30 is 29
Explanation
The prime numbers between 11 and 30 are 11, 13, 17, 19, 23, and 29.
So the greatest prime number is 29.
Problem 6
Is 97 a prime number?
Yes, 97 is a prime number.
Explanation
The factors of 97 are 1 and 97, so it is only divisible by 1 and 97.
FAQs on Prime Numbers
1.Why is 2 a prime number?
2.Are all prime numbers odd?
3.How do you calculate the prime numbers?
4.What is a co-prime?
5.Is 0 a prime number?
6.What is the smallest prime number?
7.Is 11 a prime number?
8.Is one prime or composite number?
9.What is the largest prime number?
10.Is 24 prime or a composite number?
11.Is 37 a prime number?
12.What is a prime triplet?
13.How can children in Indonesia use numbers in everyday life to understand Prime Numbers?
14.What are some fun ways kids in Indonesia can practice Prime Numbers with numbers?
15.What role do numbers and Prime Numbers play in helping children in Indonesia develop problem-solving skills?
16.How can families in Indonesia create number-rich environments to improve Prime Numbers skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
