Prime Numbers Examples: 44 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Prime 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 whose only positive divisors are 1 and themselves—they cannot be factored further.

Primes can't be broken down further—they're the 'atoms' of multiplication.

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 prime is a whole number bigger than $1$ that cannot be split into a product of smaller whole numbers.

Common stuck point: The procedure for prime numbers is the easy part; the trap is calling 1 prime. Asking "Does this number bigger than $1$ have exactly two factors — $1$ and itself — and no others?" 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 exactly two factors — $1$ and itself — and no others?

Worked Examples

Example 1

easy

Find the prime factorization of

180

Answer

2^{2} \times 3^{2} \times 5

First step

Divide by 2:

180 \div 2 = 90

. Divide again:

90 \div 2 = 45

Full solution

2
45 is odd, try 3: $45 \div 3 = 15$ . Again: $15 \div 3 = 5$ .
3
$5$ is prime, so stop. Prime factorization: $180 = 2^2 \times 3^2 \times 5$ .

To find the prime factorization, repeatedly divide by the smallest prime that divides evenly, working upward. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).

Example 2

medium

Determine whether

97

is prime.

Example 3

easy

Find the prime factorization of

60

Example 4

medium

Find the prime factorization of

360

Example 5

medium

Use prime factorization to find

\gcd(84, 90)

Example 6

medium

Find the prime factorization of

1000

Example 7

hard

Determine whether

221

is prime.

Example 8

hard

p

and

p + 2

are both prime and

p > 3

, prove that

p + 1

is divisible by

6

Example 9

hard

How many divisors does

720

have?

Example 10

challenge

Show that there are infinitely many primes (Euclid's argument).

Practice Problems

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

Example 1

easy

List all prime numbers between 20 and 40.

Example 2

easy

51

prime or composite?

Example 3

easy

7

a prime number?

Example 4

easy

1

a prime number?

Example 5

easy

2

a prime number?

Example 6

easy

9

prime or composite?

Example 7

easy

List all prime numbers less than 10.

Example 8

easy

15

prime?

Example 9

easy

What is the smallest prime number?

Example 10

easy

11

prime?

Example 11

medium

Write the prime factorization of

60

Example 12

medium

51

prime? Justify.

Example 13

medium

Find the prime factorization of

84

Example 14

medium

How many divisors does

12

have? Use its prime factorization.

Example 15

medium

Are

14

and

15

both composite? Identify their prime factors.

Example 16

medium

What is the largest prime factor of

90

Example 17

medium

Twin primes differ by 2. Find a pair of twin primes between 10 and 20.

Example 18

challenge

Why is

2

the only even prime number?

Example 19

challenge

Prove there is no largest prime (Euclid's idea, briefly).

Example 20

challenge

Find the smallest number with exactly three distinct prime factors.

Example 21

medium

Find the prime factorization of

120

Example 22

medium

97

prime? Check divisibility by primes up to its square root.

Example 23

easy

13

a prime number?

Example 24

easy

List all primes between

40

and

60

Example 25

easy

1

prime, composite, or neither?

Example 26

easy

39

prime?

Example 27

medium

143

prime?

Example 28

medium

Find two prime numbers whose sum is

24

Example 29

medium

How many primes are between

1

and

30

Example 30

medium

Twin primes are primes that differ by

2

. Find all twin prime pairs less than

50

Example 31

hard

How many distinct prime factors does

2310

have?

Example 32

hard

Find a prime

p

such that

p

p + 4

, and

p + 8

are all prime.

Example 33

hard

Determine whether

\gcd(2^{10}, 3^{10}) = 1

and explain.

Example 34

challenge

Find all primes

p

such that

p^2 + 2

is also prime.

← Back to Prime Numbers Formula Explained Practice Mode

Related Concepts

Factors Divisibility Intuition Composite Numbers Prime Factorization

Background Knowledge

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

factorsdivisibility intuition