Multiples Examples in Math

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

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

Concept Recap

Numbers obtained by multiplying a given number by positive integers: the skip-counting sequence $n, 2n, 3n, 4n, \ldots$

Skip-counting produces multiples: counting by 3s gives 3, 6, 9, 12... — those are the multiples of 3.

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: Multiples are the products you get by repeatedly adding the same number.

Common stuck point: The procedure for multiples is the easy part; the trap is listing factors instead of multiples. Asking "Can the number be written as the given number times a whole number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can the number be written as the given number times a whole number?

Worked Examples

Example 1

easy

List the first

6

multiples of

7

, and find the

20

th multiple of

7

Answer

First

6

7, 14, 21, 28, 35, 42

. The

20

th multiple is

140

First step

Multiples of

7

7 \times 1 = 7

7 \times 2 = 14

7 \times 3 = 21

7 \times 4 = 28

7 \times 5 = 35

7 \times 6 = 42

Full solution

2
First $6$ multiples: $7, 14, 21, 28, 35, 42$ .
3
$20$ th multiple: $7 \times 20 = 140$ .

Multiples of

n

are the values

n, 2n, 3n, \ldots

— the results of multiplying

n

by positive integers. The

k

th multiple is simply

kn

. There are infinitely many multiples of any non-zero integer.

Example 2

medium

Find all multiples of

6

between

50

and

100

(inclusive). How many are there?

Example 3

easy

List the first six multiples of

11

. What pattern do you see in the digits?

Example 4

medium

Find the third common multiple of

6

and

10

(besides their LCM).

Example 5

hard

A number

N

is a multiple of

6

and ends in

0

. The smallest such

N

greater than

50

is what?

Practice Problems

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

Example 1

easy

252

a multiple of

7

? Of

12

? Show your reasoning.

Example 2

medium

A light flashes every

4

seconds and a buzzer sounds every

6

seconds. They start together. When will they next coincide? How many times in one minute?

Example 3

easy

List the first four multiples of

3

Example 4

easy

20

a multiple of

5

Example 5

easy

What is the

6

th multiple of

4

Example 6

easy

List the multiples of

10

up to

50

Example 7

easy

27

a multiple of

3

Example 8

easy

Do the multiples of

7

ever stop?

Example 9

easy

Which of

14, 15, 16

is a multiple of

4

Example 10

easy

What is the smallest positive multiple of

9

Example 11

medium

Find the least common multiple of

4

and

6

Example 12

medium

What is the

10

th multiple of

7

, and is it divisible by

5

Example 13

medium

List the common multiples of

3

and

5

that are under

40

Example 14

medium

A bell rings every

6

minutes, another every

8

minutes. They ring together at noon. When next together?

Example 15

medium

How many multiples of

4

are there between

1

and

40

inclusive?

Example 16

medium

0

a multiple of

5

? Discuss the usual convention.

Example 17

medium

The multiples of

n

are

n,2n,3n,\ldots

. What is the difference between consecutive multiples?

Example 18

medium

Find the least common multiple of

8

and

12

Example 19

medium

How many multiples of

5

lie between

1

and

100

inclusive?

Example 20

challenge

Prove that the sum of two multiples of

n

is again a multiple of

n

Example 21

challenge

Show that

\mathrm{lcm}(a,b)\cdot\gcd(a,b)=a\cdot b

for

a=12,b=18

, and state the general identity.

Example 22

challenge

Prove that every multiple of

6

is also a multiple of both

2

and

3

Example 23

easy

List the first five multiples of

8

Example 24

easy

Which numbers in

\{12,15,18,22\}

are multiples of

3

Example 25

medium

How many multiples of

7

lie between

1

and

100

inclusive?

Example 26

medium

Find the least common multiple of

9

and

12

Example 27

medium

List the common multiples of

4

and

6

that are at most

50

Example 28

medium

Two bells ring together at noon. Bell A rings every

10

minutes, Bell B every

15

minutes. When do they next ring together?

Example 29

medium

How many multiples of

8

are there between

50

and

150

inclusive?

Example 30

medium

List the multiples of

13

less than

100

Example 31

medium

Three friends jog around a track. They finish a lap every

4,6,

and

8

minutes. If they start together, when do they all finish a lap at the same time again?

Example 32

hard

How many multiples of

6

are there from

1

200

inclusive that are NOT multiples of

4

Example 33

hard

Find the smallest number greater than

1000

that is a multiple of both

25

and

35

Example 34

hard

A multiple of

7

between

200

and

300

has digits that sum to

9

. Find it.

Example 35

hard

Two integers have LCM

60

and one is

12

. What are the possible values of the other?

Example 36

hard

How many integers from

1

1000

are multiples of

3

5

Example 37

hard

Find all multiples of

7

between

1

and

200

whose digits sum to a multiple of

7

Example 38

hard

Show that the sum of any three consecutive multiples of

5

is divisible by

15

Example 39

challenge

Find the smallest positive integer that is a multiple of

2,3,4,5,6,7,8,9,

and

10

Example 40

challenge

How many multiples of

4

from

1

500

are NOT multiples of

6

← Back to Multiples Formula Explained Practice Mode

Related Concepts

Multiplication Least Common Multiple Divisibility Intuition

Background Knowledge

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

multiplication