# All About Prime Numbers: Definition, Properties, 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?

83

61

71

81

2)What is the 10th prime number?

23

29

31

37

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

1

2

3

4

4)Which is the smallest odd prime number?

1

2

3

5

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.