Math Academy

Finding Remainders

Introduction

Writing Division Problems With Remainders as a Multiplication Problem

Finding a Remainder

Finding a Remainder Given an Equivalent Multiplication Problem

Finding the Remainder of a Division Problem Given the Quotient

Prerequisites

The Relationship Between Multiplication and Division

Understanding Remainders Using Models

Evaluating Expressions Involving Addition and Multiplication

Introduction

Let's consider the following division problem model:

This model represents the division problem

We can express this division as a multiplication problem. However, as well as multiplying, we must add the remainder:

We can also swap the order of the multiplication as follows:

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Example: Writing Division Problems With Remainders as a Multiplication Problem

Consider the following division problem:

Which of the following problems are equivalent to our division problem?

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

We can swap the order of the multiplication, as follows:

Therefore, the correct answer is "I and III only."

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Practice: Writing Division Problems With Remainders as a Multiplication Problem

Consider the following division problem: $33 \div 4 = 8 \, \textrm{R} \: 1$

Which of the following problems are equivalent to our division problem?

$33 = 4\times 1 + 8$
$33 = 8\times 4 + 1$
$33 = 4\times 8 + 1$

a	I only
b	III only
c	II and III only
d	I and III only
e	II only

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

$33 = 4\times 8 + {\color{blue}{1}}$

We can swap the order of the multiplication, as follows:

$33 = 8\times 4 + {\color{blue}{1}}$

Therefore, the correct answer is "II and III only."

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Practice: Writing Division Problems With Remainders as a Multiplication Problem

Consider the following division problem: $26 \div 7 = 3 \, \textrm{R} \: 5$

Which of the following problems are equivalent to our division problem?

$26 = 7\times 3 + 5$
$26 = 3\times 7 + 5$
$26 = 7\times 5 + 3$

a	III only
b	II only
c	I and II only
d	I only
e	I and III only

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

$26 = 7\times 3 + {\color{blue}{5}}$

We can swap the order of the multiplication, as follows:

$26 = 3\times 7 + {\color{blue}{5}}$

Therefore, the correct answer is "I and II only."

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Finding a Remainder

How would we find the remainder in the division problem below?

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

Now, since the above equation becomes

So, the remainder must be

Finally, we can write

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Example: Finding a Remainder Given an Equivalent Multiplication Problem

Consider the following division problem with an unknown remainder.

We can rewrite this division problem as follows:

What is the unknown remainder?

EXPLANATION

Since our equation becomes

So, the remainder must be

Finally, we can write

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Practice: Finding a Remainder Given an Equivalent Multiplication Problem

Consider the following division problem with an unknown remainder. $29 \div 10 = 2 \, \textrm{R} \: \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$ We can rewrite this division problem as follows:

a
b
c
d
e

EXPLANATION

Since $10 \times 2 = 20,$ our equation becomes: $29 = 20\,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{9}}.$ $29 = 20\,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}9}$

Finally, we can write $29 \div 10 = 2 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}9}.$

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Practice: Finding a Remainder Given an Equivalent Multiplication Problem

Consider the following division problem with an unknown remainder. $29 \div 4 = 7 \, \textrm{R} \: \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

We can rewrite this division problem as follows:

$29 = 4\times 7 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

What is the unknown remainder?

a	$5$
b	$2$
c	$1$
d	$3$
e	$0$

EXPLANATION

Since $4\times 7 = 28,$ our equation becomes:

$29 = 28 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{1}}.$

$29 = 28 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}1}$

Finally, we can write

$29 \div 4 = 7 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}1}.$

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Practice: Finding a Remainder Given an Equivalent Multiplication Problem

Consider the following division problem with an unknown remainder. $70\div 9 = 7 \, \textrm{R} \: \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$ We can rewrite this division problem as follows:

a
b
c
d
e

EXPLANATION

Since $9 \times 7 = 63,$ our equation becomes: $70 = 63 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{7}}.$ $70 = 63 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}7}$

Finally, we can write $70 \div 9 = 7 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}7}.$

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Example: Finding the Remainder of a Division Problem Given the Quotient

Find the remainder in the division problem below.

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

Now, since the above equation becomes

So, the remainder must be

Finally, we can write

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Practice: Finding the Remainder of a Division Problem Given the Quotient

Find the remainder in the division problem below.

a
b
c
d
e

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder. $44 = 10 \times 4 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

Now, since $10 \times 4 = 40,$ the above equation becomes: $44 = 40\,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{4}}.$ $44 = 40\,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}4}$

Finally, we can write $44 \div 10 = 4 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}4}.$

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Practice: Finding the Remainder of a Division Problem Given the Quotient

Find the remainder in the division problem below.

a
b
c
d
e

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder. $17 = 3 \times 5 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

Now, since $3 \times 5 = 15,$ the above equation becomes: $17 = 15 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{2}}.$ $17 = 15\,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}2}$

Finally, we can write $17 \div 3 = 5 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}2}.$

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Practice: Finding the Remainder of a Division Problem Given the Quotient

Find the remainder in the division problem below. $91 \div 8 = 11 \, \textrm{R} \: \fbox{$\phantom{A}$}$

a	$3$
b	$1$
c	$11$
d	$2$
e	$8$

EXPLANATION

First, we express our division as a multiplication problem. However, as well as multiplying, we add the remainder.

$91 = 8\times 11 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

Now, since $8\times 11 = 88,$ the above equation becomes:

$91 = 88 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}?}$

So, the remainder must be ${\color{blue}{3}}.$

$91 = 88 \,\,+\,\, \bbox[3pt, white, border: 1px solid black]{\color{blue}3}$

Finally, we can write

$91 \div 8 = 11 \, \textrm{R} \, \bbox[3pt, white, border: 1px solid black]{\color{blue}3}.$

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