Composite Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Composite Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Integers greater than 1 that can be expressed as a product of two smaller positive integers; they are the opposite of primes.

Numbers that can be built by multiplying smaller numbers together.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A composite is a whole number bigger than $1$ that factors into smaller whole numbers.

Common stuck point: The procedure for composite numbers is the easy part; the trap is calling 1 composite. Asking "Does this number bigger than $1$ have at least one factor other than $1$ and itself?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this number bigger than $1$ have at least one factor other than $1$ and itself?

Worked Examples

Example 1

easy

Determine whether

91

is prime or composite. If composite, find a factor pair.

Answer

91

is composite:

91 = 7 \times 13

First step

Test divisibility by primes up to

\sqrt{91} \approx 9.5

: primes to test are

2, 3, 5, 7

Full solution

2
$91$ is odd (not divisible by $2$ ). Digit sum $= 10$ (not divisible by $3$ ). Last digit $\neq 0, 5$ (not by $5$ ).
3
Test $7$ : $91 \div 7 = 13$ . Yes! $91 = 7 \times 13$ .
4
$91$ is composite with factor pair $(7, 13)$ .

To test primality, check all primes up to

\sqrt{n}

. If none divide

n

, it is prime; if any divide

n

, it is composite.

91

is a classic 'looks prime' trap — many students guess prime because neither

7

nor

13

are obvious factors.

Example 2

medium

List all composite numbers between

20

and

35

, and for each, give one non-trivial factor pair.

Example 3

medium

Find all factor pairs of

36

to show it is composite.

Example 4

medium

List all composite numbers between

50

and

60

, with one factor pair each.

Example 5

medium

You have

24

tiles. List all rectangles (length

\times

width) you can build with them. Why does this show

24

is composite?

Example 6

hard

Find three consecutive composite numbers.

Example 7

hard

Show that

n! + 2, n! + 3, \ldots, n! + n

are all composite (for

n \ge 2

Example 8

hard

A number

n

leaves remainder

0

when divided by

3

and by

5

. List the four smallest composite values of

n

greater than

1

Example 9

challenge

For

n \ge 4

, prove that

n^4 + 4

is composite for every integer

n \ge 2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

1

composite? Is

4

composite? Is

2

composite? Explain each briefly.

Example 2

medium

How many composite numbers are there from

1

20

? List them and explain why each is composite.

Example 3

easy

9

a composite number?

Example 4

easy

7

prime or composite?

Example 5

easy

1

prime, composite, or neither?

Example 6

easy

Give a composite number between

10

and

15

Example 7

easy

15

composite? Show a factorization.

Example 8

easy

Which is composite:

2,3,4,5

Example 9

easy

25

composite?

Example 10

easy

How many factors does a prime number have, versus a composite?

Example 11

medium

Find the prime factorization of

12

Example 12

medium

Show

12

has more than two prime factors counted with repetition.

Example 13

medium

List all composite numbers between

1

and

10

Example 14

medium

Find the smallest odd composite number.

Example 15

medium

How many divisors does

12

have? Use its prime factorization.

Example 16

medium

51

prime or composite? Test divisibility.

Example 17

medium

Why is every even number greater than

2

composite?

Example 18

medium

Find the prime factorization of

60

Example 19

medium

How many divisors does

36

have? Use its prime factorization.

Example 20

challenge

Prove that every composite number

n

has a prime factor

\le\sqrt{n}

Example 21

challenge

Show that

n^2-1

is composite for every integer

n>2

Example 22

challenge

Prove that the product of two consecutive integers greater than

1

is composite.

Example 23

easy

21

composite? If yes, give a factor pair other than

1 \times 21

Example 24

easy

Which is composite:

11

13

, or

14

Example 25

easy

How many composite numbers are between

1

and

10

(inclusive)?

Example 26

easy

Give two different composite numbers between

30

and

40

Example 27

easy

True or false: every even number greater than

2

is composite.

Example 28

medium

How many composite numbers are between

40

and

60

inclusive?

Example 29

medium

143

prime or composite? Justify with a factorization if composite.

Example 30

medium

Find the largest composite number less than

50

Example 31

medium

169

composite?

Example 32

medium

How many factors does the composite number

36

have?

Example 33

medium

Give a composite number that is also a perfect square between

50

and

100

Example 34

hard

What is the smallest composite number greater than

100

Example 35

hard

Determine whether

221

is prime or composite.

Example 36

hard

1001

prime or composite?

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Related Concepts

Prime Numbers Factors Prime Factorization Greatest Common Factor

Background Knowledge

These ideas may be useful before you work through the harder examples.

prime numbersfactors