Composite Numbers: Definition, Examples, Properties and Tricks

Composite Numbers are positive integers greater than 1 that have more than two factors. This page explains their factor properties, prime factorisation, divisor-count formula, identification methods and common exceptions. Worked examples include even and odd composite numbers, the special cases of 1 and 0, and methods for solving composite number questions quickly.

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What Are Composite Numbers?

A composite number is a positive integer greater than 1 that has more than two positive factors. Equivalently, it can be written as a product of two smaller positive integers.

For example, 12 has factors 1, 2, 3, 4, 6 and 12, so it is composite. The smallest composite number is 4 because its factors are 1, 2 and 4. The number 1 is neither prime nor composite, while 2 and 3 are prime numbers.

Composite Numbers Video Lessons

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Types of Numbers Classification

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Composite Numbers Formula & Tricks

Important Formulas

Number of divisors
If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then d(n) = (a₁ + 1)(a₂ + 1) ... (aₖ + 1)

A number is composite when its divisor count d(n) is greater than 2.

Composite numbers up to n
Number of composite integers from 1 to n = n − 1 − π(n)

Here, π(n) represents the number of primes less than or equal to n. The subtraction excludes 1 and all prime numbers.

Composite-number test
If n > 1 and n = a × b, where 1 < a < n and 1 < b < n, then n is composite.

Finding any proper factor other than 1 and the number itself proves that the number is composite.

Quick Tricks

Check divisors only up to the square root

To decide whether n is composite, test divisibility by prime numbers up to √n. If one divides n exactly, n is composite; if none does, n is prime.

Example: For 91, √91 is about 9.54. Since 91 is divisible by 7, it is composite: 91 = 7 × 13.
Use the even-number rule

Every even integer greater than 2 is composite because it has 2 as a proper factor.

Example: 28 is composite because 28 = 2 × 14. The number 2 itself is prime, so it is the only even prime.
Apply quick divisibility tests

A number divisible by 3, 5, 7 or another divisor greater than 1 and less than the number is composite. For 3, add the digits; for 5, check whether the last digit is 0 or 5.

Example: 123 has digit sum 6, so it is divisible by 3. Therefore, 123 is composite: 123 = 3 × 41.

Composite Numbers Concepts

Factor Count and Composite Numbers

A number is composite when it has at least three positive divisors.

Every integer greater than 1 has at least the divisors 1 and itself. A third divisor indicates that the number has a factor pair other than 1 and the number itself. For example, 18 has six divisors: 1, 2, 3, 6, 9 and 18.

Example: The divisors of 15 are 1, 3, 5 and 15. Since it has four divisors, 15 is composite.

Prime Factorisation Test

Prime factorisation shows whether a number is composite and gives its total number of divisors.

If a number is expressed as a product of two or more primes, it is composite. For n = 2³ × 3², the divisor count is (3 + 1)(2 + 1) = 12, so n has more than two divisors and is composite.

Example: 72 = 2³ × 3². Therefore, d(72) = (3 + 1)(2 + 1) = 12.

Prime, Composite and Neither

A prime number has exactly two positive factors, a composite number has more than two, and 1 has exactly one positive factor.

The number 1 is neither prime nor composite because it has only one factor. The first prime numbers are 2, 3, 5 and 7. The first composite numbers are 4, 6, 8, 9 and 10.

Example: 7 has only 1 and 7 as factors, so it is prime. 9 has 1, 3 and 9 as factors, so it is composite.

Testing a Number for Compositeness

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

A composite number n has at least one factor not exceeding √n. Thus, checking divisibility by primes up to √n is sufficient. For 97, √97 is less than 10, and 97 is not divisible by 2, 3, 5 or 7; hence 97 is prime, not composite.

Example: For 119, √119 is about 10.9. Since 119 is divisible by 7, 119 = 7 × 17, so it is composite.

Counting Composite Numbers in a Range

The count of composite numbers from 1 to n is n minus 1 and the number of primes up to n.

The integers from 1 to n contain one number that is neither prime nor composite, namely 1. Every remaining number is either prime or composite. Therefore, subtract 1 and the prime count π(n) from n.

Example: From 1 to 10, π(10) = 4 because the primes are 2, 3, 5 and 7. Composite count = 10 − 1 − 4 = 5: 4, 6, 8, 9 and 10.

Composite Numbers Data Table

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Classification of Small Integers

Examples of prime, composite and neither-prime-nor-composite numbers.

Number Factors Classification
1 1 Neither prime nor composite
2 1, 2 Prime
3 1, 3 Prime
4 1, 2, 4 Composite
9 1, 3, 9 Composite
12 1, 2, 3, 4, 6, 12 Composite

Note: A prime number has exactly two factors; a composite number has more than two factors.

Practice Composite Numbers Questions

Practise published questions related to this topic.

1√ 90.25 + √ 6,25 - √ 2.25 + √ 0.25 = ?→ 2The value of ( √(42 + √(( 46 + √(( 3 + √(( 34 + √4 )) )) ))) ) is:→ 3Which is the greatest number that divides 47, 59 and 74 to leave the same remainder?→ 4Two numbers have HCF 14 and LCM 420. If their difference is 98, find the numbers.→ 5If the cost price of 12 identical books is ₹960 and the cost price of 8 identical pens is ₹320, find the total cost price (in ₹) of 15 books and 10 pens of the same kind.→ 6The ratio of two numbers is 3:4 and their HCF is 7. Their LCM is:→ 7The number 640500 is not divisible by:→ 8If the HCF of three numbers is 23 and they are in the ratio 1:2:3, then the numbers are:→ 9The product of two prime numbers is 119. The numbers are:→ 10If 15 notebooks cost ₹240, then how many notebooks can be bought with ₹992?→

Composite Numbers Quick Quiz

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Quick Revision Notes

Composite Numbers: Quick Revision

Recall the definitions, tests and formulas used to identify and count composite numbers.

  • A composite number is an integer greater than 1 with more than two positive factors.
  • The smallest composite number is 4.
  • The number 1 is neither prime nor composite.
  • Every even number greater than 2 is composite.
  • If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then d(n) = (a₁ + 1)(a₂ + 1) ... (aₖ + 1).
  • A number is composite if it has a divisor between 2 and n − 1.
  • For a compositeness test, check prime divisors only up to √n.
  • Composite numbers from 1 to n = n − 1 − π(n).

Related Topic Explainers

Number System → Prime Numbers → Divisibility Rules → Divisibility Rule of 2 → Divisibility Rule of 3 → Divisibility Rule of 4 → Divisibility Rule of 6 → Divisibility Rule of 8 →

Composite Numbers FAQs

What is the smallest composite number?

The smallest composite number is 4. Its positive factors are 1, 2 and 4.

Is 1 a composite number?

No. The number 1 is neither prime nor composite because it has only one positive factor.

Is every odd number composite?

No. Some odd numbers are prime, such as 3, 5 and 7. Odd numbers such as 9, 15 and 21 are composite.

How many factors does a composite number have?

A composite number has more than two positive factors. The exact count can be found from its prime factorisation using d(n) = (a₁ + 1)(a₂ + 1) ... (aₖ + 1).

Why is 2 not composite?

2 has exactly two positive factors, 1 and 2, so it is prime. It is the only even prime number.

How can 91 be shown to be composite?

Since 91 = 7 × 13, it has proper factors 7 and 13. Therefore, 91 is composite.

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