Factors Examples: 45 Problems with Answers

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

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

Concept Recap

Whole numbers that divide evenly into a given number with no remainder—the 'building blocks' that multiply together to make it.

Factors are the 'building blocks' you multiply together to make a number.

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 factor is a whole-number piece in a multiplication pair.

Common stuck point: The procedure for factors is the easy part; the trap is listing multiples when asked for factors. Asking "Does this number multiply with another whole number to make the target?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this number multiply with another whole number to make the target?

Worked Examples

Example 1

easy

List all factors of

56

Answer

1, 2, 4, 7, 8, 14, 28, 56

First step

Test divisors from 1 up to

\sqrt{56} \approx 7.5

56 \div 1 = 56

56 \div 2 = 28

56 \div 4 = 14

56 \div 7 = 8

Full solution

2
$56 \div 3$ , $56 \div 5$ , and $56 \div 6$ are not whole numbers.
3
Factors: $\{1, 2, 4, 7, 8, 14, 28, 56\}$ .

To find all factors, test integers from 1 up to

\sqrt{n}

. Each divisor

d

that works gives a pair:

d

and

\frac{n}{d}

. This guarantees you find every factor.

Example 2

medium

How many factors does

72

have?

Example 3

easy

List all factors of

36

Example 4

medium

List all factors of

100

Example 5

medium

Use divisibility rules to determine whether

1428

is divisible by

12

Example 6

hard

How many factors does

1024

have?

Example 7

hard

How many odd factors does

360

have?

Example 8

hard

Find a number less than

50

that has the most factors.

Example 9

challenge

For how many positive integers

n \le 100

does

n

have an odd number of factors?

Practice Problems

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

Example 1

easy

132

divisible by

6

? Explain using divisibility rules.

Example 2

easy

List all factor pairs of

36

Example 3

easy

List all the factors of

12

Example 4

easy

4

a factor of

20

Example 5

easy

Find the factor pair of

24

that includes

3

Example 6

easy

List all factors of

7

Example 7

easy

5

a factor or a multiple of

10

Example 8

easy

How many factors does

1

have?

Example 9

easy

List all factors of

16

Example 10

easy

6

a factor of

30

Example 11

medium

List all factors of

36

Example 12

medium

How many factors does

24

have? Use its prime factorization.

Example 13

medium

Find the common factors of

18

and

24

Example 14

medium

What is the sum of all factors of

12

Example 15

medium

n

is a factor of 18 and also a factor of 27, list all possible

n

Example 16

medium

Which number between 20 and 30 has the most factors?

Example 17

medium

13

a factor of

52

? Show the check.

Example 18

challenge

Why does a perfect square like

36

have an odd number of factors?

Example 19

challenge

A number has exactly 3 factors. What kind of number must it be?

Example 20

challenge

Find the smallest number greater than 1 that has exactly 6 factors.

Example 21

medium

Find all common factors of

24

and

36

Example 22

medium

How many factors does

48

have?

Example 23

easy

List all factors of

18

Example 24

easy

List all factor pairs of

24

Example 25

easy

How many factors does

10

have?

Example 26

easy

9

a factor of

54

Example 27

medium

How many factors does

48

have?

Example 28

medium

Find the sum of all factors of

20

Example 29

medium

How many factor pairs does

48

have?

Example 30

medium

Find a number less than

30

with exactly

6

factors.

Example 31

medium

List the common factors of

30

and

45

Example 32

hard

Find the smallest positive integer with exactly

12

factors.

Example 33

hard

How many factors of

144

are perfect squares?

Example 34

hard

Sum the factors of

28

. Is

28

a perfect number?

Example 35

hard

How many factors of

2025

are odd?

Example 36

challenge

Find the smallest positive integer divisible by every integer from

1

10

← Back to Factors Formula Explained Practice Mode

Related Concepts

Divisibility Intuition Multiplication Prime Numbers Greatest Common Factor

Background Knowledge

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

divisibility intuitionmultiplication