Prime Numbers: Definition, Chart, Examples, and Practice Problems

What is a Prime Number

A prime number is a natural positive number that is only divisible by 1 and the number itself. To rephrase it, prime numbers are positive integers that are greater than 1 with only 2 factors, 1 and the number itself. The numbers 2, 3, 5, 7, 11, 13 are a few examples of prime numbers. All other numbers are composite numbers. The only exception is 1 which is neither a prime nor composite number. This is because its value is not greater than 1 and it is not a product of two smaller natural numbers. All prime numbers are odd numbers except for 2 which is the smallest prime number and the only even prime number. A prime number is only divisible by 1 and its own self.

Properties and Facts of Prime Numbers

Prime numbers are the natural numbers which are greater than 1.
2 is the smallest prime number.
2 is the only even prime number. Every other prime number is an odd number.
Any two prime numbers are always coprime.
Any composite number can be expressed as the product of its prime factors.
A prime number has only two factors – 1 and the number itself.
If 2 prime numbers have only 1 composite number between them, they are known as twin-prime numbers.
TThe only consecutive prime numbers are 2 and 3.
Except for 0 and 1, a whole number is either a prime or a composite number.
There is no prime number greater than 5 that ends with a 5.
The Sieve of Eratosthenes is one of the earliest and easiest methods of finding prime numbers.
Prime numbers get more rare as the number gets bigger.
Every even positive integer greater than 2 can be written as the sum of two prime numbers.
Every integer greater than 1 has at least one prime factor.
The largest known prime number is 282,589,933 – 1, with 24,862,048 digits as of November 2022.

The Sieve of Eratosthenes

The Sieve of Eratosthenes is a simple method for identifying prime numbers devised in the third century B.C. by Eratosthenes, the Greek mathematician.

Step 1: Make a table of all the natural numbers between 1 to 100 using 10 rows and 10 columns.

Step 2: Cut out 1 as it is neither a prime nor composite number.

Step 3: Encircle 2 and cross out all its multiples like 4, 6, 8, 10 and so on as they are not prime numbers.

Step 4: Circle 3 which is the next uncrossed number and crosses out all its multiples. Leave out the previously crossed out numbers like 6, 12 and 18 and so on.

Step 5: Continue the process of circling the next uncrossed number and crossing out its multiples till all the numbers in the table are either encircled or crossed except 1.

Prime Number Chart

This prime numbers list shows the 25 prime numbers between 1 and 100 — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Co-Primes and Twin-Primes

Two numbers are said to be co-primes when 1 is their only common factor. Co-prime numbers are not necessarily prime numbers. For example, 9 and 10 are co-primes. Note that any two prime numbers are always coprime. This is because out of their two factors, 1 is the only common factor. Some examples of co-prime numbers are (3, 5), (11,13), (17, 19) and so on.

Twin-prime numbers are 2 prime numbers that have just one composite number between them. (3, 5), (5, 7), (11, 13), (17, 19) and so on are some examples of twin-prime numbers.

Practice Problems of Prime Numbers with Solutions

1)Identify the given numbers as prime numbers or composite numbers.

13, 48, 49, 23, 74, 80, 71, 59, 45, 47

Solution:

Prime Numbers	Composite Numbers
13, 23, 71, 59, 47	48, 49, 74, 80, 45

2) Express the number 21 as the sum of two prime numbers.

Solution:

We can write 21 as

21=19+2

Here, 2 and 19 are both prime numbers.

3) Which are the prime numbers between 20 and 30?

Solution:

The 2 prime numbers between 20 and 30 are 23 and 29.

4) Which is the greatest prime number between 80 and 90?

Solution:

The prime numbers between 80 and 90 are 83 and 89 which makes

89 the greatest prime number.

5) Is 10 a prime number?

Solution:

No, because it can be divided evenly by 2 or 5, 2×5=10, as well as by 1 and 10.

Alternatively, you can use the formula of 6n ± 1.

10 = 6(1) + 4 = 6(2) – 2

This is not of the form 6n + 1 or 6n – 1.

Hence, 10 is not a prime number.

6) Is 19 a prime number?

Solution:

Write the number in the form of 6n ± 1.

6(3) + 1 = 18 + 1 = 19

Therefore, 19 is a prime number.

7) Which is the greatest prime number between 1 to 10?

Solution:

The 4 prime numbers between 1 and 10 are 2, 3, 5, and 7 so the greatest prime number is 7.

8) What is the largest known prime number?

Solution:

The largest known prime number is 282,589,933 – 1, with 24,862,048 digits as of November 2022.

Practice Problems on Prime Numbers

1)Which of the following is not a prime number?

2)What is the 10th prime number?

3)How many prime numbers are there between 40 and 50?

4)Which is the smallest odd prime number?

5)Which of the following pairs of numbers are co-prime numbers?

(7, 14)

(15, 27)

(25, 35)

(8, 55)

Frequently Asked Questions on Prime Numbers

Is 1 a prime number?
Can a prime number be negative?
Why is 2 the only even prime number?
Which is the largest known prime number?
In which way are prime numbers and co-prime numbers different?
Which is the lowest prime number?
What makes a number prime?
How to find prime numbers?
How do you check if a number is a prime number?
What is the smallest prime number?
Why are 0 and 1 not considered to be prime nor composite numbers?
What is the total number of prime numbers from 0 to 100?

With this blog, we hope you have more clarity on prime numbers, their properties, methods to find prime numbers and different prime numbers lists. There are also a few examples and practice problems with and without solutions based on these concepts for a better understanding. For more clarity on prime numbers and their various properties and features, you can visit the EuroKids website and update yourself.