Prime And Composite Numbers — 3–5 Math | Scholars & Parents

What Is This?

Every whole number greater than 1 falls into one of two categories based on how many factors it has:

A prime number has exactly two factors: 1 and itself.
A composite number has more than two factors — it can be divided exactly by at least one number other than 1 and itself.

Special case: The number 1 is neither prime nor composite. It has only one factor (itself).

Why Does It Matter?

Prime numbers are the building blocks of all whole numbers. Every composite number can be broken down (factorised) into a unique product of prime numbers. This underpins simplifying fractions, finding GCFs and LCMs, and forms the basis of modern cryptography and computer security.

Prime Numbers

Definition

A prime number has exactly two factors: 1 and itself.

Examples

Number	Factors	Prime?
2	1, 2	✓ Prime
3	1, 3	✓ Prime
5	1, 5	✓ Prime
7	1, 7	✓ Prime
11	1, 11	✓ Prime
13	1, 13	✓ Prime

Key Facts About Primes

2 is the only even prime number. Every other even number is divisible by 2, so it has at least three factors.
The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
There are infinitely many prime numbers.

Composite Numbers

Definition

A composite number has three or more factors — it can be divided by at least one number other than 1 and itself.

Examples

Number	Factors	Composite?
4	1, 2, 4	✓ Composite
6	1, 2, 3, 6	✓ Composite
9	1, 3, 9	✓ Composite
12	1, 2, 3, 4, 6, 12	✓ Composite
15	1, 3, 5, 15	✓ Composite

The Number 1 — Special Case

1 is neither prime nor composite.

It has only one factor (1 itself).
Primes need exactly two factors; composites need three or more.
1 fits neither definition.

The Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient method for finding all prime numbers up to a given limit. Here is how to find all primes up to 50:

Step 1: Write all numbers from 2 to 50.

Step 2: Circle 2 (the first prime). Cross out all multiples of 2 (4, 6, 8, …).

Step 3: Circle 3 (the next uncrossed number). Cross out all multiples of 3 (6, 9, 12, …).

Step 4: Circle 5. Cross out multiples of 5.

Step 5: Circle 7. Cross out multiples of 7.

Step 6: All remaining uncrossed numbers are prime.

Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Prime Factorisation

Every composite number can be written as a product of prime numbers. This is called its prime factorisation.

Method — Factor Trees

Example: Prime factorisation of 36

        36
       /  \
      4    9
     / \  / \
    2   2 3   3

36 = 2 × 2 × 3 × 3 = 2² × 3²

Example: Prime factorisation of 60

        60
       /  \
      6    10
     / \  /  \
    2   3 2    5

60 = 2 × 3 × 2 × 5 = 2² × 3 × 5

Key Property

No matter how you build the factor tree, the prime factorisation of a number is always the same — this is called the Fundamental Theorem of Arithmetic.

Testing Whether a Number Is Prime

To check whether a number n is prime, test whether any prime number up to √n divides it exactly.

Example: Is 37 prime?

√37 ≈ 6.1, so test primes up to 6: 2, 3, 5.
37 ÷ 2 = 18.5 ✗
37 ÷ 3 = 12.33 ✗
37 ÷ 5 = 7.4 ✗
No prime divides 37 → 37 is prime ✓

Example: Is 51 prime?

√51 ≈ 7.1, so test primes up to 7: 2, 3, 5, 7.
51 ÷ 3 = 17 exactly ✓
51 has a factor other than 1 and itself → 51 is composite (51 = 3 × 17)

Worked Examples

Worked Example 1: Classify 29

Factors: 1 and 29 only (29 ÷ 2, 3, 5 are all non-whole) → prime ✓

Worked Example 2: Classify 49

49 ÷ 7 = 7 → factors include 1, 7, 49 → composite (49 = 7²) ✓

Worked Example 3: Prime factorisation of 48

48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3
48 = 2⁴ × 3 ✓

Worked Example 4: Use prime factorisation to find GCF(12, 18)

12 = 2² × 3
18 = 2 × 3²
Common prime factors: 2¹ × 3¹ = 6 ✓

Common Mistakes

Mistake 1: Thinking 1 is prime 1 has only one factor. A prime requires exactly two. 1 is neither prime nor composite.

Mistake 2: Thinking all odd numbers are prime 9 = 3 × 3, 15 = 3 × 5, 25 = 5 × 5, 27 = 3³ — all odd, all composite.

Mistake 3: Forgetting that 2 is prime 2 is the only even prime. Students sometimes dismiss it because "even = composite," but 2 has exactly two factors (1 and 2).

Mistake 4: Stopping a factor tree too early 48 = 6 × 8, but 6 and 8 are not prime. Keep splitting until every branch ends in a prime number.

Key Takeaways

Prime: exactly two factors (1 and itself) — e.g. 2, 3, 5, 7, 11, 13 …
Composite: three or more factors — e.g. 4, 6, 8, 9, 10, 12 …
1 is neither prime nor composite
2 is the only even prime number
Every composite number has a unique prime factorisation (factor tree)
To test primality: divide by primes up to the square root

Practice and Resources

Ready to practise? Try our Grade 3-5 Prime and Composite Numbers Worksheet covering classification, the Sieve, factor trees, and prime factorisation. Test yourself with the Grade 3-5 Math Practice Test.