266 LearnersLast updated on August 29, 2025
Prime Numbers
Prime numbers are natural numbers greater than one. They are only divisible by one and the number itself. We use prime numbers in cryptography, internet security, digital signatures, and so on.
Importance of Prime Numbers in Mathematics
Based on their properties, numbers are classified into different types. The numbers are classified into prime and composite numbers based on the divisibility. In mathematics, prime numbers are used as the building blocks of all positive integers. That is, all positive composite numbers can be expressed as products of prime numbers.
Key Concepts Related to Prime Numbers
Prime numbers form the basis of many mathematical concepts. In this section, let’s discuss the key concepts related to prime numbers.
Prime and Composite Numbers
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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 Example: 11 is only divisible by 1 and 11 |
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
The pair of prime numbers where the difference between the prime numbers is 2 are the twin primes. In between a pair of twin primes, there is a composite number.
For example, (3, 5), (5, 7), (11, 13), (17, 19), etc.
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), etc.
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 all the odd numbers are not 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 consider a large number to be prime without checking 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 the divisibility till the square root of 257. √257 ≈ 16.03, so check divisibility up to 16.
Mistake 3
Thinking Negative Numbers are Prime
Prime numbers are defined only for positive integers greater than 1, so negative numbers like -2, -3, -5 are not prime.
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 the number should be prime and the difference between them should be 2.
Mistake 5
Misunderstanding the concept of prime and co-primes.
Confusing between co-primes and prime numbers is a common mistake. Students think that co-primes should be prime numbers, but they are wrong. Co-primes can be any number where the greatest common factor should be 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 it. These tricks can help you learn prime numbers quickly.
All the even numbers except 2 are non-prime numbers because it has more than 2 factors.
For example, 256 is not a prime number as it has more than two factors. Any two distinct prime numbers are always co-prime because their greatest common divisor (GCD) is 1.
- 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 5 as it is divisible by 5.
- The numbers ending in 0 are not prime numbers as it is divisible by 2 and 10.
- A famous unproven conjecture (Goldbach's conjecture) suggests that every even number greater than 2 can be expressed as the sum of two prime numbers. For example; 10 = 5 + 5; 8 = 5 + 3; 16 = 11 + 5.
Real-world Applications of Prime Numbers
In our daily life, we use prime numbers in different fields such as cryptography and data security, analyzing number patterns, and so on. Let’s learn a few real-world applications of prime numbers.
- Cryptography and Data Security: Prime numbers are used to encrypt algorithms through public-key cryptography like RSA. Cryptography is used to decrypt or encrypt data. For data security, prime numbers can be used as passwords.
- Simplification of fraction: Prime numbers can be used to find the GCF of more than two numbers. GCF is used to simplify the fractions.
- Analyzing number pattern: As prime numbers are considered the building blocks of all integers, it helps to understand the relationship between numbers. So we use prime numbers in hash tables to spread 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.How can children in Saudi Arabia use numbers in everyday life to understand Prime Numbers?
8.What are some fun ways kids in Saudi Arabia can practice Prime Numbers with numbers?
9.What role do numbers and Prime Numbers play in helping children in Saudi Arabia develop problem-solving skills?
10.How can families in Saudi Arabia 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.
