How to Identify a Prime Number: Step-by-Step with Examples

How to Identify a Prime Number is an important topic that helps students understand the basics of numbers. A prime number has only two factors, 1 and itself, which makes it easy to test a number in a simple way. Learning this topic helps students build strong number skills and solve maths questions with more confidence.

Table of Contents

• What is a Prime Number?
• What is NOT a Prime Number?
• Prime vs Composite Numbers
• Properties of Prime Numbers
• How to Identify a Prime Number: Simple Methods
• Special Types of Prime Numbers
• Prime Numbers List
• Solved Examples
• Practice Questions

What is a Prime Number?

A prime number is a number that:

• Is greater than 1

• Can only be divided exactly by 1 and itself

• Has no other factors

Examples of Prime Numbers

Number 7:

Can 7 be divided by 1? Yes (7 ÷ 1 = 7) 

Can 7 be divided by 7? Yes (7 ÷ 7 = 1) 

Can 7 be divided by 2? No (gives remainder)

Can 7 be divided by 3? No (gives remainder)

Can 7 be divided by 4, 5, 6? No

So 7 is a PRIME NUMBER!

Number 6:

Can 6 be divided by 1? Yes 

Can 6 be divided by 2? Yes (6 ÷ 2 = 3) 

Can 6 be divided by 3? Yes (6 ÷ 3 = 2) 

Can 6 be divided by 6? Yes 

6 has MORE than two factors: NOT a prime number.

What is NOT a Prime Number?

Number 1:

Many students think 1 is prime. It's NOT. Because 1 only has one factor (itself). Prime numbers must have exactly two factors.

Even numbers (except 2):

All even numbers are divisible by 2, so they have more than two factors. None of them are prime except 2 itself

Numbers with obvious factors:

Numbers ending in 0 or 5 (except 5 itself) are always divisible by 5, so they're never prime.

Prime vs Composite Numbers

Properties of Prime Numbers

Understanding these properties helps you identify prime numbers faster.

Property 1: Smallest Prime Number

The smallest prime number is 2

2 is also the ONLY even prime number!

Because every other even number is divisible by 2, giving it an extra factor.

Property 2: Only Even Prime

2 is the only even prime number

All other even numbers are composite

(they're all divisible by 2)

Property 3: All Other Primes are Odd

Every prime number greater than 2 is an odd number

Examples: 3, 5, 7, 11, 13, 17, 19, 23...

But remember: Not all odd numbers are prime!

Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15...

9 = 3 × 3 (not prime)

15 = 3 × 5 (not prime)

Property 4: Infinitely Many Primes

Prime numbers go on forever. There's no biggest prime number. Mathematicians have discovered prime numbers with millions of digits!

Property 5: Every Number Has a Prime Factorization

Every whole number greater than 1 can be written as a product of prime numbers.

12 = 2 × 2 × 3

30 = 2 × 3 × 5

100 = 2 × 2 × 5 × 5

How to Identify a Prime Number Simple Methods

Now let's learn the actual methods for checking if a number is prime.

Method 1: Division Method (Basic Method)

This is the simplest way to check if a number is prime.

Step 1: Check if the number is less than 2

If yes: NOT prime

If no: Continue checking

Step 2: Check if the number equals 2

If yes: It IS prime (2 is prime)

Step 3: Check if the number is even

If yes: NOT prime (all even numbers except 2 are composite)

Step 4: Divide the number by odd numbers starting from 3

If any division gives no remainder: NOT prime

If no division gives an exact result: It Is prime

Example: Is 17 prime?

Step 1: 17 > 2 (continue)

Step 2: 17 ≠ 2 (continue)

Step 3: 17 is odd (continue)

Step 4: Divide by odd numbers:

   17 ÷ 3 = 5.67 (not exact)

   17 ÷ 5 = 3.4 (not exact)

   17 ÷ 7 = 2.43 (not exact)

   No exact divisions found!

17 IS a prime number!

Example: Is 21 prime?

Step 1: 21 > 2

Step 2: 21 ≠ 2

Step 3: 21 is odd (continue)

Step 4: Divide by odd numbers:

21 ÷ 3 = 7 (exact No remainder)

21 is NOT prime (it equals 3 × 7)

Method 2: Square Root Method (Faster Method)

The Rule: To check if n is prime, only divide by numbers

up to the square root of n (√n)

If none of them divide n exactly n is prime.

If n has a factor larger than its square root, it must also have a factor smaller than its square root. So checking up to √n is enough!

Example: Is 37 prime?

Step 1: Find √37 ≈ 6.08

Step 2: Check all primes up to 6: (2, 3, 5)

37 ÷ 2 = 18.5 (not exact)

37 ÷ 3 = 12.33 (not exact)

37 ÷ 5 = 7.4 (not exact)

No exact divisions! 37 is prime.

Example: Is 49 prime?

Step 1: Find √49 = 7

Step 2: Check primes up to 7: (2, 3, 5, 7)

49 ÷ 2 = 24.5 (not exact)

49 ÷ 3 = 16.33 (not exact)

49 ÷ 5 = 9.8 (not exact)

49 ÷ 7 = 7 (exact No remainder)

49 = 7 × 7 NOT prime.

Method 3: Sieve of Eratosthenes

This ancient method (invented by a Greek mathematician) helps find ALL prime numbers up to a certain value at once!

Steps to find all primes up to 50:

Step 1: Write all numbers from 2 to 50

2  3  4  5  6  7  8  9  10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

Step 2: Start with 2. Cross out all multiples of 2 (except 2 itself)

Cross out: 4, 6, 8, 10, 12, 14, 16, 18, 20...

Step 3: Move to 3. Cross out all multiples of 3 (except 3 itself)

Cross out: 6, 9, 12, 15, 18, 21, 24...

Step 4: Move to 5. Cross out all multiples of 5

Cross out: 10, 15, 20, 25, 30, 35...

Step 5: Move to 7. Cross out all multiples of 7

Cross out: 14, 21, 28, 35, 42, 49...

Step 6: Numbers NOT crossed out are PRIME!

Primes up to 50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Elimination Table:

Prime Numbers List

Prime Numbers from 1 to 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, 97

Total prime numbers from 1 to 100 = 25

Prime Numbers Table (1 to 100)

Prime Numbers from 1 to 200

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,

41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,

89, 97, 101, 103, 107, 109, 113, 127, 131,

137, 139, 149, 151, 157, 163, 167, 173, 179,

181, 191, 193, 197, 199

Special Types of Prime Numbers

Twin Primes:

Twin primes are pairs of prime numbers that differ by 2.

Examples of twin prime pairs:

(3, 5)

(5, 7)

(11, 13)

(17, 19)

(29, 31)

(41, 43)

(59, 61)

(71, 73)

Co-Prime Numbers:

Two numbers are co-prime when their only common factor is 1.

Examples:

(8, 15) Factors of 8: 1, 2, 4, 8

          Factors of 15: 1, 3, 5, 15

          Common factor: only 1 Co-prime.

(4, 9) Co-prime.

(7, 11) Co-prime. (both prime)

Note: Two prime numbers are always co-prime to each other!

Mersenne Primes

Primes of the form 2ⁿ - 1 (where n is also prime).

2² - 1 = 3 (prime!)

2³ - 1 = 7 (prime!)

2⁵ - 1 = 31 (prime!)

2⁷ - 1 = 127 (prime!)

Solved Examples

Example 1: Basic Prime Check

Question: Is 29 a prime number?

Solution:

Step 1: Check basic rules

29 is odd

29 doesn't end in 0 or 5

Digit sum: 2 + 9 = 11 (not divisible by 3)

Step 2: Find square root

√29 ≈ 5.4

Check primes up to 5: (2, 3, 5)

Step 3: Divide

29 ÷ 2 = 14.5 (not exact)

29 ÷ 3 = 9.67 (not exact)

29 ÷ 5 = 5.8 (not exact)

Answer: 29 is a prime number.

Example 2: Composite Check

Question: Is 91 a prime number?

Solution:

Step 1: Check basic rules

91 is odd

Digit sum: 9 + 1 = 10 (not divisible by 3)

Doesn't end in 0 or 5

Step 2: Find square root

√91 ≈ 9.5

Check primes up to 9: (2, 3, 5, 7)

Step 3: Divide

91 ÷ 2 = 45.5 (not exact)

91 ÷ 3 = 30.33 (not exact)

91 ÷ 5 = 18.2 (not exact)

91 ÷ 7 = 13 (exact! No remainder)

Answer: 91 is NOT a prime number (91 = 7 × 13).

Example 3: Finding Primes in a Range

Question: Find all prime numbers between 20 and 40.

Solution:

Check each odd number (even ones aren't prime):

21: 2 + 1 = 3 (divisible by 3) NOT prime

23: √23 ≈ 4.8 Check 2, 3

23 ÷ 2 = 11.5 (not exact)

23 ÷ 3 = 7.67 (not exact)

23 is prime.

25: Ends in 5 NOT prime (25 = 5 × 5)

27: 2 + 7 = 9 (divisible by 3) NOT prime

29: √29 ≈ 5.4 Check 2, 3, 5

29 ÷ 2 = 14.5 (not exact)

29 ÷ 3 = 9.67 (not exact)

29 ÷ 5 = 5.8 (not exact)

29 is prime.

31: √31 ≈ 5.6 Check 2, 3, 5

31 ÷ 2 = 15.5 (not exact)

31 ÷ 3 = 10.33 (not exact)

31 ÷ 5 = 6.2 (not exact)

31 is prime.

33: 3 + 3 = 6 (divisible by 3) NOT prime

35: Ends in 5 NOT prime

37: √37 ≈ 6.1 Check 2, 3, 5

37 ÷ 2 = 18.5 (not exact)

37 ÷ 3 = 12.33 (not exact)

37 ÷ 5 = 7.4 (not exact)

37 is prime.

39: 3 + 9 = 12 (divisible by 3) NOT prime

Answer: Prime numbers between 20 and 40 are 23, 29, 31, 37.

Example 4: Using Sieve of Eratosthenes

Question: Find all prime numbers between 1 and 30.

Solution:

Step 1: Write numbers 2 to 30

2  3  4  5  6  7  8  9  10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Step 2: Keep 2, cross out multiples of 2

Cross out: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30

Step 3: Keep 3, cross out multiples of 3

Cross out: 9, 15, 21, 27 (others already crossed)

Step 4: Keep 5, cross out multiples of 5

Cross out: 25 (others already crossed)

Step 5: Keep 7 (√30 ≈ 5.5, so we stop here)

Remaining numbers (primes):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Answer: Prime numbers from 1 to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Example 5: Twin Prime Identification

Question: Find all twin prime pairs between 1 and 50.

Solution:

First list all primes from 1 to 50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Now check which pairs differ by exactly 2:

(3, 5) → 5 - 3 = 2 Twin prime pair.

(5, 7) → 7 - 5 = 2 Twin prime pair

(11, 13) → 13 - 11 = 2 Twin prime pair.

(17, 19) → 19 - 17 = 2 Twin prime pair.

(29, 31) → 31 - 29 = 2 Twin prime pair.

(41, 43) → 43 - 41 = 2 Twin prime pair.

Answer: Twin prime pairs from 1 to 50 are (3,5), (5,7), (11,13), (17,19), (29,31), (41,43).

Practice Questions

Section 1: Yes or No, Check whether each number is prime:

Question 1: Is 2 a prime number?

Question 2: Is 1 a prime number?

Question 3: Is 13 a prime number?

Question 4: Is 27 a prime number?

Question 5: Is 47 a prime number?

Question 6: Is 51 a prime number?

Question 7: Is 71 a prime number?

Question 8: Is 91 a prime number?

Question 9: Is 97 a prime number?

Question 10: Is 100 a prime number?

Section 2: Fill in the Blanks

Question 11: The smallest prime number is ___.

Question 12: The only even prime number is ___.

Question 13: The prime numbers between 10 and 20 are ___, ___, ___, ___.

Question 14: The number 1 is neither prime nor ___.

Question 15: Two prime numbers that differ by 2 are called ___ primes.

Section 3: Find the Primes

Question 16: List all prime numbers from 40 to 60.

Question 17: How many prime numbers are there between 1 and 50?

Question 18: Find all twin prime pairs between 50 and 100.

Question 19: Which of these are prime: 63, 67, 69, 71, 77?

Question 20: Find the prime numbers in this list: 15, 17, 21, 23, 25, 29, 33, 37.