Notebook

00:05

28 Mar 2026

Using Euclid's Division Lemma

Welcome! Now that you've mastered the theory, we're going to put Euclid's Division Lemma into action and start solving problems.

We'll begin with two primary problem types:

The Direct Find: You're given the divisor, quotient, and remainder and must find the original number.
The Mystery Divisor: The dividend is known, but the divisor is the unknown value you need to solve for.

To solve these, we will practice rearranging the equation:

$a = bq + r$ Lemma

Unknown	Action
Dividend ( $a$ )	Direct calculation
Divisor ( $b$ )	Algebraic rearrangement

Finally, we'll encounter a special case where the algebra tells us 'this is impossible'.

Recognizing that signal is just as important as solving the problems that work!

1. Find the dividend

We're moving from understanding Euclid's Division Lemma to actually using it.

The first type of problem gives you the divisor, quotient, and remainder, and asks you to find the original number (the dividend).

How to Solve:

Read the problem carefully
Assign each value to the right slot in the formula: $a = bq + r$ formula
Compute the answer
Check your work

Goal: Use the lemma to find the unknown dividend ( $a$ ).

📋 Given Info

Euclid's Division Lemma

Remember the core formula:

$a = bq + r$

Where the condition for the remainder is:

0 \leq r < b

key rule

Key Terms:

$a$ = Dividend (the number being divided)
$b$ = Divisor
$q$ = Quotient
$r$ = Remainder

✍️ Question

Problem ✏️

A number when divided by 47 gives 28 as quotient and 13 as remainder.

Find the number. Show your working.

Step 1: Identify the Divisor

First, let's read the problem and assign each value based on Euclid's Division Lemma formula:
$a = b q + r$

'Divided by 47' $\to$ $b = 47$

Note: The number mentioned after the phrase "divided by" is always the divisor, represented by the letter $b$ in our formula.

✍️ MCQ

Choose one

If the problem says 'the quotient is 28', which letter does 28 represent?

Step 2: Assign Quotient and Remainder

Now, we extract the remaining numerical values from the problem statement:

'Quotient is 28' $\to$ $q = 28$
(The letter $q$ stands for quotient)
'Remainder is 13' $\to$ $r = 13$
(The letter $r$ stands for remainder)

We now have our values for $b$ , $q$ , and $r$ .

✍️ MCQ

Choose one

Which variable represents the unknown 'Dividend' we need to find?

Step 3: Identify the Unknown

'A number' $\to$ $a = ?$ find this!When the question asks for a number, it means the unknown dividend

The problem asks us to find the "number" that is being divided. In our lemma, this is the dividend, represented by the variable $a$ The dividend is the starting number, always labeled as a.

This is our unknown value that we are ready to solve for!

Calculating the Dividend ( $a$ )

To find the value of $a$ using Euclid's Division LemmaA formal method to verify division results, we apply the formula:

$a = b q + r$

Given values:

Divisor ( $b$ ) = $47$
Quotient ( $q$ ) = $28$
Remainder ( $r$ ) = $13$ The remainder is added last, after multiplication

Equation:

a = 47 \times 28 + 13

(Multiply divisor and quotient first)

Note: To solve this correctly, we must follow the order of operationsMust complete multiplication before adding remainder (BODMAS/PEMDAS) by performing the multiplication before the addition.

✍️ MCQ

Choose one

In the expression

47 \times 28 + 13

, which operation do we perform first?

Step 1: Multiplication ( $b \times q$ )

To make the multiplication $47 \times 28$ simpler, we can use the distributive propertyHelps avoid calculation errors with large numbers by splitting $28$ into $(20 + 8)$ .

First Part:

$47 \times 20$ Break numbers into easier parts to multiply
Mental Math Tip: Calculate $47 \times 2 = 94$ , then append a zero.
Result: $940$

✍️ FIB

Fill in the blank

What is

47 \times 8

Step 2: Completing the Calculation

Second Part:

$47 \times 8$
Break it down: $(40 \times 8) + (7 \times 8)$
$320 + 56$ = $376$

Combine for the total product ( $b q$ ):

$940 + 376$ = $\mathbf{1316}$ b×qTotal value before adding the remainder

This represents the final product of the divisor and the quotientThe product before accounting for the remainder.

Calculating the Dividend

Now, let's complete the final step of the division formula:

Product of Divisor and Quotient: $47 \times 28 = 1316$ Your starting point for building the dividend
Add the Remainder ( $r = 13$ ):

$$a = 1316 + 13 = \mathbf{1329}$$$

The final value of the dividend $a$ is 1329What you get after adding the remainder.

✍️ MCQ

Choose one

What does the value

a = 1329

represent?

Verifying the Result

Whenever you apply Euclid's Division Lemma, you must verify the remainder constraintMost common place where students make mistakes:

The Rule

For $a = b q + r$ , it must always be true that:

0 \le r < b

(Always check this condition)

Quick Check:

Is $0 \leq 13 < 47$ ?
Yes! Since the remainder ( $13$ ) is non-negative and smaller than the divisor ( $47$ ), the constraint is satisfiedProves your division follows Euclid's Lemma correctly.

Final Answer: The calculation is valid, and the result is 1329.

✍️ Yes/No

Yes or No?

Is a remainder of

50

valid if the divisor is

47

2. Find the divisor

The Problem Flips! 🔄

You've already found the dividend by computing $a = b q + r$ .

Now the problem flips:

The dividend ( $a$ ) is known
The divisor ( $b$ ) is the mystery

The equation is the same, but this time you need to rearrange it to find what you're looking for.

📋 Given Info

What you know:

Euclid's Division Lemma:

$a = bq + r$

The Remainder Condition:

$0 \leq r < b$ key rule

Where:

$a$ is the Dividend
$b$ is the Divisor
$q$ is the Quotient
$r$ is the Remainder

✍️ Question

Your Turn ✏️

By what number should 1,365 be divided to get 31 as quotient and 32 as remainder?

Find the number.
Show your working.

1. Mapping the Problem to Variables

Let's break down the word problem by assigning each value to our variables in the context of the division algorithm:

Dividend ( $a$ )The number being divided: The number being divided is $1365$ .
Divisor ( $b$ )What you divide by: This is our unknown value ("by what number").
Quotient ( $q$ )One of your results: We are given that the result of the division is $31$ .
Remainder ( $r$ )(The leftover value): We are given a leftover value of $32$ .

Variable	Value
$a$ (Dividend)	$1365$ The dividend value
$b$ (Divisor)	$?$ find thisThe unknown we need to find
$q$ (Quotient)	$31$
$r$ (Remainder)	$32$

✍️ MCQ

Choose one

If the remainder

r = 32

, what must be true about the divisor

b

2. Setting Up the Equation

Using Euclid's Division Lemma ( $a = b q + r$ ), we substitute our known values to create an equation to solve for $b$ :

$1365 = b (31) + 32$

Important Condition: Recall that in any division, the remainder $r$ must be less than the divisor $b$ ( $0 \leq r < b$ ). Since our remainder is $32$ , we know for certain that our answer $b$ must be greater than $32$ The remainder is always smaller than the divisor.

✍️ MCQ

Choose one

What is the first step to solve for

b

1365 = 31b + 32

Algebra is Algebra!

Don't let these larger numbers intimidate you. This is simply a linear equation in $b$ A fixed order of operations to follow, following the exact same logic you've used since Class 7.

Comparing the Logic

Class 7 Example	Current Problem
$3 x + 5 = 20$	$31 b + 32 = 1365$
Step 1: Subtract $5$ to isolate the $x$ term.Isolate the variable term before dividing	Step 1: Subtract $32$ to isolate the $b$ term.
Step 2: Divide by $3$ to find $x$ .	Step 2: Divide by $31$ to find $b$ .

Key takeaway: The numbers are bigger, but the method remains identical.Same method you have already learned

✍️ MCQ

Choose one

In the equation

31b + 32 = 1365

, which term should be moved (subtracted) first?

Solving for $b$

Applying those same Class 7 steps to our problem:

Equation:
$31 b + 32 = 1365$

Subtract 32 from both sides:
$31 b = 1365 - 32$
$31b = 1333$ Simple balance scale to solve
Divide by 31 to isolate $b$ :
$b = \frac{1333}{31}$ To find our value of b
$b = 43$

Even with three-digit or four-digit numbers, the algebraic steps are exactly the same.Same approach for small or large numbers

✍️ Yes/No

Yes or No?

Is the calculated divisor

b = 43

greater than the remainder

r = 32

Step 1 — Subtract 32 from both sides: $1365 - 32 = 31 b$ $1333 = 31 b$

Step 2 — Divide both sides by 31: $b = \frac{1333}{31}$

Step 3 — Long division:

Therefore $b = 43$ .

Verification Step

Quick check: Is $0 \leq 32 < 43$ ?

Yes, the constraint is satisfied.

The Remainder Rule: Always verify that the remainder ( $r$ ) is smaller than the divisor ( $b$ )This is the most critical part of Euclid's Lemma.

Since our remainder 32 is indeed less than our divisor 43, the condition $0 \leq r < b$ Always double check this condition in exams is satisfied and our answer is correct.

✍️ Yes/No

Yes or No?

If we had calculated a divisor

b = 30

and our remainder was

r = 32

, would that answer be valid?

A Universal Approach

This method works for solving any unknown variable in Euclid's Division Lemma.

Write the Lemma: Start with $a = bq + r$ Start with the standard form.
Identify the Unknown: Mark whether you are looking for the dividend ( $a$ ), divisor ( $b$ ), quotient ( $q$ ), or remainder ( $r$ ).
Isolate the Variable: Use basic algebra to rearrange the equationThe method never changes.

Key Insight: Whether you are finding $b$ , $q$ , or $r$ Regardless of which part is missing, the logic remains identicalThis consistency is key — isolate the variable using algebraic manipulation.

✍️ MCQ

Choose one

To isolate

r

in the equation

a = bq + r

, what is the first step?

3. Recognising an impossible problem

So far every problem has given a clean whole-number answer. But what if someone gives you values that don't actually work together?

Recognising when the algebra is telling you 'this is impossible' is just as important as solving the ones that work.

You know Euclid's Division Lemma:

$a = bq + r$

Where the remainder condition is:

$0 \leq r < b$
key constraint

Let's see if you can spot when something doesn't add up.

✍️ Question

Problem 🔍

When 25 is divided by 4, the remainder is claimed to be 3.

Find the quotient.

Problem Setup: Division Parameters

Let's work through this problem step by step. We are given the following values based on the statement "When 25 is divided by 4, the remainder is 3":

Dividend ( $a$ ): 25aThe number we begin with
Divisor ( $b$ ): 4bHow big each group is
Remainder ( $r$ ): 3rWhat's left after making groups

Goal: Our objective is to determine the value of the quotient ( $q$ )How many complete groups we can make.

✍️ MCQ

Choose one

In the statement '25 is divided by 4', which number represents the divisor

b

Applying Euclid's Division Lemma

We can represent this division using the formal structure of Euclid's Division LemmaThe key formula for this chapter:

$a = bq + r$

(The structure of the formula)

Substituting our known values into the formula:

$25 = 4q + 3$

(Plugging in our numbers)

This equation aligns the given numbers into the $bq + r$ structureMatching the formula pattern, allowing us to isolate and solve for the unknown quotient $q$ .

✍️ MCQ

Choose one

What is the first step to solve for

q

25 = 4q + 3

Solving for the Quotient ( $q$ )

To isolate the term with $q$ , we subtract $3$ from both sides of the equation:

$25 - 3 = 4 q$ $22 = 4 q$

Now we are just one step away from finding the value of $q$ .

✍️ MCQ

Choose one

22 = 4q

, what is the value of

q

Calculating $q$

Dividing both sides by $4$ , we get:

$q = \frac{22}{4} = 5.5$

Wait! Look closely at this result. $5.5$ is a decimal, not a whole number.

Recall the Rules of EDL

In Euclid's Division Lemma ( $a = b q + r$ ), both the quotient $q$ and the remainder $r$ must be whole numbers.

✍️ Yes/No

Yes or No?

Can the quotient

q

be a decimal in Euclid's Division Lemma?

Conclusion: An Impossible Case

Since $q$ must be a whole number in Euclid's Division Lemma, a decimal result like $5.5$ is not allowed. This leads us to a key realization:

The given combination is impossibleA non-integer means it's impossible.
The math simply doesn't add upCheck if the division and remainder are possible because you cannot divide $25$ by $4$ and get a remainder of $3$ .

Rule: If solving for $q$ or $r$ results in a non-integer valueviolationNon-integer values violate the lemma, the scenario violates the lemma.

Verifying the Division

Let's double-check the process with actual division to see how the numbers interact:

Multiplication: $4 \times 6 = 24$
Subtraction: $25 - 24 = 1$

Here, 1remainderThe specific amount left over after taking maximum groups is our actual remainder. It is the amount left over after we have taken the maximum number of groups of 4 from 25.

✍️ MCQ

Choose one

In the division

25 = 4 \times 6 + 1

, what is the quotient

q

The Correct Division Equation

Based on our calculation, we can express the relationship between these numbers using Euclid's Division Lemma structure:

$25 = 4 \times 6 + 1$ The only way these integers can relate under the division lemma

Key Observations:

The real remainderIf you get a decimal quotient, your division setup was incorrect is 1, not 3.
This explains why earlier attempts to force a different remainder led to impossible decimal quotients; the numbers must fit this specific integer relationship.

✍️ Yes/No

Yes or No?

Does the remainder

r = 1

satisfy the condition

0 \leq r < b

where the divisor

b = 4

💡 Key Insight: Dealing with Non-Whole Numbers

When your rearrangement or calculation gives a non-whole-number answer (like $5.5$ ) for $q$ or $r$ , it is a massive red flagDecimals signal something went wrong.

Finding a decimal usually means one of two things:

Arithmetic Error:Check your math for mistakes You likely made a mistake in your calculation (like subtraction or division).
Flawed Problem:The question may be impossible The problem itself might be impossible or incorrectly stated.

Always ensure your quotient and remainder remain within the set of whole numbersOnly whole numbers are valid.

✍️ Yes/No

Yes or No?

Can the quotient

q

or the remainder

r

be a decimal value?

📏 The Whole Number Rule

In Euclid's Division Lemma, the quotient $q$ and the remainder $r$ must always be whole numbersZero, one, two, and so on ( $0, 1, 2, . . .$ ).

(key rule)If they aren't, the division isn't valid!(Invalid division means something's wrong)

If you get a decimal or fraction: Stop immediately!
Action step: Double-check your subtraction and division steps first.Look for simple calculation mistakes You may have missed a simple calculation step.

Using Euclid's Division Lemma

1. Find the dividend

How to Solve:

Euclid's Division Lemma

Problem ✏️

Step 1: Identify the Divisor

Step 2: Assign Quotient and Remainder

Step 3: Identify the Unknown

Calculating the Dividend (aaa)

Step 1: Multiplication (b×qb \times qb×q)

Step 2: Completing the Calculation

Calculating the Dividend

Verifying the Result

The Rule

2. Find the divisor

The Problem Flips! 🔄

What you know:

Your Turn ✏️

1. Mapping the Problem to Variables

2. Setting Up the Equation

Algebra is Algebra!

Comparing the Logic

Solving for bbb

Verification Step

A Universal Approach

3. Recognising an impossible problem

Problem 🔍

Problem Setup: Division Parameters

Applying Euclid's Division Lemma

Solving for the Quotient (qqq)

Calculating qqq

Recall the Rules of EDL

Conclusion: An Impossible Case

Verifying the Division

The Correct Division Equation

💡 Key Insight: Dealing with Non-Whole Numbers

📏 The Whole Number Rule

Calculating the Dividend ( $a$ )

Step 1: Multiplication ( $b \times q$ )

Solving for $b$

Solving for the Quotient ( $q$ )

Calculating $q$