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00:09

28 Mar 2026

Factors, Multiples, and How EDL Splits Integers into Categories

Welcome! Today we are diving into Factors, Multiples, and How EDL Splits Integers into Categories. We are going to see how simple division can actually help us organize every number in existence.

What happens when the division comes out perfectly — no leftover at all?

The remainder is $0$ .

That simple fact connects to two words you will see everywhere from now on:

Factor
Multiple

But $r = 0$ is just one of the possible remainders. For any divisor $b$ , there are exactly $b$ possible remainders:

Divisor ( $b$ )	Possible Remainders ( $r$ )
$2$	$0, 1$
$3$	$0, 1, 2$
$b$	$0, 1, . . ., b - 1$

This means every positive integer falls into one of $b$ neat, non-overlapping categories.

That sorting power is what makes proofs about even and odd numbers — and much more — possible.

1. Factors and multiples from r = 0

You've been using Euclid's Division Lemma to solve problems where the remainder $r$ is always some positive number like $1, 2, 13,$ or $32$ .

But what about when the division comes out perfectly, with nothing left over?

The remainder is 0 ( $r = 0$ )
This connects to two key terms: factor and multiple

When the remainder is zero, the divisor is a factor of the dividend, and the dividend is a multiple of the divisor.

📋 Given Info

You know Euclid's Division Lemma:

$a = b q + r, where 0 \leq r < b$

Consider this decomposition:

$63 = 7 \times 9 + 0$

The remainder $r$ is $0$ .

Key Terms:

When $r = 0$ , we say:

$b$ is a 'factor' of $a$
$a$ is a 'multiple' of $b$

✍️ Question

Question 🤔

In the decomposition $63 = 7 \times 9 + 0$ :

Is 7 a factor or a multiple of 63?
Is 63 a factor or a multiple of 7?

Case: Zero Remainder ( $r = 0$ )

When the remainder $r = 0$ , Euclid's Division Lemma equation simplifies to:

$a = b q$

In this special scenario, the divisor ( $b$ )The divisor fits perfectly into the dividend fits into the dividend ( $a$ )No gap left when dividing perfectly, like a puzzle piece, with absolutely nothing left overZero remainder means nothing left over. This signifies a perfect division relationship between the two numbers.

✍️ Yes/No

Yes or No?

r = 0

in the equation

a = bq + r

, does it mean that

b

divides

a

exactly?

The Relationship: Factors and Multiples

When division is perfect ( $a = b q$ ), we use two specific mathematical terms to describe the relationship between the numbers:

FactorTwo perspectives on the same relationship: We say that $b$ is a factor of $a$ (it divides $a$ exactly).
MultipleTwo perspectives on the same relationship: We say that $a$ is a multiple of $b$ (it is the product of $b$ and some integer $q$ ).

Example: In the equation $63 = 7 \times 9$ , we can precisely say that $7$ is a factor of $63$ Seven divides sixty-three exactly, and $63$ is a multiple of $7$ Sixty-three is the product of seven times nine.

✍️ MCQ

Choose one

12 = 3 \times 4

, which word best describes the number

12

in relation to

3

Understanding Factors

Definition

A factorA number that divides evenly into another is a number that divides another number exactlyDivides another number exactly, leaving no leftover partsThe remainder must be exactly zero (remainder is zero).

Example

7 divides 63 exactly: $7 \times 9 = 63 + 0$ Since the remainder is $0$ , 7 is a factor of 63.

Key Rule: A factor must fit perfectly into the larger number without any remainderCheck the remainder—it must be zero.

✍️ Yes/No

Yes or No?

15 \div 4

leaves a remainder of

3

, is

4

a factor of

15

Think of it as: The smaller number that 'goes into' the bigger one.

Other Factors of 63

In addition to 7, the factors of 63 include:

1, 3, 9, 21, and 63 itself.

Important Observations

A number is always a factor of itselfEvery number divides itself exactly once: This is because every number divides itself exactly one timeNo remainder when dividing by itself.
Universal Factor: 1 is a factor of every number.

✍️ MCQ

Choose one

Why is

3

a factor of

63

What is a Multiple?

Multiple: The result you get when you multiply a number by a whole numberThe result of multiplying by a whole number (an integer).

Since $63 = 7 \times 9$ , we say that 63 is a multiple of 763 is what you get from 7 times 9. It is the productresultThe finished product of multiplication of that multiplication.

If factors are the "building blocks," multiples are the finished product.

✍️ MCQ

Choose one

4 \times 5 = 20

, which of these is the multiple of

5

Multiples of 7

Think of a multiple as the bigger numberMultiples are always the larger number that is "made from" the smaller one. You can find multiples by skip-countingUse skip-counting to find multiples or looking at a number's multiplication table.

Multiples of 7: $7, 14, 21, 28, 35, 42, 49, 56, 63, 70, \dots$

[!TIP] Key Insight: Every number is a multiple of itself!A number is always its own multiple Notice that the list starts with 7starts with 7The list begins with the number itself, just as 7 is also a factor of 7.

✍️ FIB

Fill in the blank

What is the next multiple of

7

after

70

The Shortcut to Remembering

Mixing up factors and multiples is a common struggle. Use this secret shortcut to keep them straight:

"Factors go intoFactors are the smaller building blocks, multiples come out."

Think of it like a recipe:

Factors are like the ingredientsThey go into the number that go into a recipe.
Multiples are the big resultsAlways equal to or larger than the original that come out of the process.

✍️ MCQ

Choose one

If a number "goes into" another exactly, it is a:

Applying the Shortcut

Let's look at the number $63$ to see how this works in practice:

The Factor ( $7$ ): Since $7$ goes into(It divides evenly) $63$ exactly ( $63 \div 7 = 9$ ), $7$ is a factorfactorWhen a number divides evenly.
The Multiple ( $63$ ): Because $63$ comes outThe product of multiplication when we multiply $7 \times 9$ , $63$ is a multiplemultipleWhen it's the result of multiplying of $7$ .

✍️ FIB

Fill in the blank

40

comes out of

8 \times 5

, then

40

is a ___ of

8

Equivalent Phrasings in Division

In mathematics, different terms are often used to describe the same relationship. Understanding these synonyms is key to decoding word problems.

1. "Divides"

Phrase: '7 divides 63(The word divides signals a factor relationship)'
Meaning: This is a shortcutDivides is shorthand for factor for saying 7 is a factor of 63.
Relationship: It implies that 7 goes into 63 perfectlyNo remainder when dividing, without any remainder.

✍️ MCQ

Choose one

If we say '

9

divides

45

', which of these is true?

2. "Divisible By"

Phrase: '63 is divisible by 7'
Meaning: This is exactly the same as saying 63 is a multiple of 7Divisible by means it's a multiple.
Relationship: If you can divide a number perfectlyClean division with no leftover by another, it means the first number "comes out" of a multiplication table. Therefore, "is divisible by" is simply the flip sideAnother way to express the same relationship of being a multiple.

✍️ Yes/No

Yes or No?

50

is a multiple of

10

, can we say '

50

is divisible by

10

The Secret of the Zero Remainder

All these terms—factor, multiple, divides, and divisible—stem from a single condition in the Euclid's Division Lemma (EDL) equation:

$r = 0$
key(The critical condition for factors and multiples)

Whenever the remainder is zeroZero remainder confirms the relationship in the EDL decomposition ( $a = b q + r$ ):

There is a perfect relationshipFactor and multiple connection exists between the divisor and the dividend.
The connection is clear: nothing is left over.

Note: If there is any remainder ( $r > 0$ ), this factor-multiple connection breaksAny leftover destroys the relationship.

✍️ Yes/No

Yes or No?

If the remainder

r

a = bq + r

4

, do we have a factor-multiple relationship?

2. Counting possible remainders

You just saw that $r = 0$ means exact divisibility. But $r$ can also be $1, 2, 3,$ and so on.

A Key Question

For a given divisor, how many different remainder values are even possible?

The answer turns out to be surprisingly simple — and it leads to something powerful.

You know the constraint from Euclid's Division Lemma:

$0 \leq r < b$

Let's see if you can use this to figure something out.

✍️ Question

Question 🤔

When you divide any positive integer by 5, how many possible values can the remainder take?

List them all.

The Remainder Constraint

In Euclid's Division Lemma, the constraint $0 \leq r < 5$ The remainder begins at zero, never negative determines the possible range for the remainder ( $r$ ).

Lower Bound: It must be at least $0$ Non-negative means zero or positive (non-negative).
Upper Bound: It must be strictly less than 5It stops right before five (the divisor).

The only whole numbers satisfying this rule are:

r = 0, 1, 2, 3, 4

(Four is the largest possible remainder when dividing by five)

Important: The remainder $r$ can never be equal to the divisor ( $5$ ), because if it were, you could divide one more time!

✍️ Yes/No

Yes or No?

If we are dividing by

5

, can the remainder be

5

The Remainder-Divisor Relationship

Notice the pattern when we count the possible values for $r$ :

$r = 0$
$r = 1$
$r = 2$
$r = 3$
$r = 4$

There are exactly 5 possible valuesFive remainders when dividing by five.

Key Insight: The number of possible remainders is always exactly equalA reliable rule to check your work to the divisor itself. If you divide by $5$ , you create exactly $5$ distinct categories5 categoriesThe divisor tells you the number of categories of numbers based on their remainders.

✍️ MCQ

Choose one

If the divisor is

3

, how many possible values can the remainder

r

have?

Two Crucial Remainder Rules

There are two specific spots where students often get tripped up when identifying possible remainders.

1. $r = 0$ is included

Zero is a valid remainderZero counts as a valid remainder value and serves as the starting pointIt's where all EDL remainder ranges begin of our remainder range ( $0 \leq r$ ).

Meaning: If $r = 0$ , the division is exactNothing left over means perfect division—there is nothing left over.
Example: $15 \div 5 = 3$ with a remainder of $0$ exactThe divisor goes in evenly.

✍️ FIB

Fill in the blank

If a division results in a remainder of

0

, we say the division is ____.

2. $r$ can never equal the divisor

If we are dividing by $5$ , the remainder can neverIt can never equal the divisor itself actually be $5$ .

The Logic: In Euclid's Division Lemma, the inequality is strict: $r < b$ Remainder is always strictly less.

Why is this the case?

If you are dividing by $5$ and find you have $5$ left over, you can simply make one more group of 5You can always fit in one more group!
If you can still fit another group in, your division isn't actually finishedThe division isn't complete yet.

Summary: For a divisor $b$ , $r$ is always less than $b$ ( $r < b$ ), never equal to it ( $r \neq b$ ).

✍️ FIB

Fill in the blank

If you are dividing a number by

6

and you find that your 'remainder' is

6

, how many more groups of

6

can you still fit in?

The Universal Remainder Rule

This pattern works for any divisor $b$ The key constraint to remember for exams: dividing by $b$ gives exactly $b$ possible remaindersThis is the constraint you must remember:

$r \in {0, 1, 2, \dots, b - 1}$

Key Points:

We always start counting remainders from zerostartNot one, always zero.
The largest possible remainder is always one less than the divisorCommon place where students make mistakes ( $b - 1$ ).

✍️ MCQ

Choose one

If the divisor is

7

, what are the possible remainders?

The Integer "Sorting Machine"Categorizes every integer by remainder

The divisor acts like a sorting machine, telling us exactly how many categories every positive integer can fall into based on its remainder:

Divisor ( $b$ )	Possible Remainders ( $r$ )	Number of Categories	Insights
1	$0$	1	$1$ divides everything; no leftover possible.
2even/oddEvery number is either even or odd	$0, 1$	2	Categorizes numbers as Even ( $0$ )Used to prove properties in board exams or Odd ( $1$ )Used to prove properties in board exams.
3	$0, 1, 2$	3	Splits all integers into $3$ neat groups.
4	$0, 1, 2, 3$	4	$4$ distinct remainder categories.
5	$0, 1, 2, 3, 4$	5	$5$ distinct remainder categories.
6	$0, 1, 2, 3, 4, 5$	6	$6$ distinct remainder categories.

Summary: The divisor tells you exactly how many categories every positive integer can fall into.

✍️ MCQ

Choose one

If a number has a remainder of

0

when divided by

2

, it is:

3. Every integer is even or odd

We just saw that dividing by $2$ gives only two possible remainders:

$r = 0$
$r = 1$

This means every positive integer falls into one of two categories.

This is something you have always known — but now you can prove it, using EDL.

✍️ Question

Question 🤔

Using Euclid's Division Lemma with a divisor of 2, explain:

Why is every positive integer either even or odd?
Why can no integer be both?

The Logic of Categories

Let's follow the chain of reasoning to see how numbers are categorized. Suppose we take any positive integerEvery number works with this format, which we will call $n$ .

If we apply Euclid's Division Lemma (EDL)Every number has a unique relationship with its divisor and remainder to this arbitrary number $n$ using a divisor of $b = 2$ , our equation becomes:

n = 2q + r

(This is the specific format for any integer)

✍️ MCQ

Choose one

When we divide by

b = 2

, what is the constraint for

r

Constraints on the Remainder

According to EDL, the remainder $r$ must satisfy the condition $0 \leq r < b$ . Since our divisor $b$ is $2$ :

The mathematical constraint is $0 \leq r < 2$ This constraint limits what remainders are possible.
The only possible whole numbers in this range are $r = 0$ or $r = 1$ only optionsWhen dividing by two, only zero or one are possible.

Key takeaway: There are no other possible values for $r$ when dividing by 2.There is no third option for any number

✍️ MCQ

Choose one

r = 0

when dividing by

2

, the number is:

Case 1: When the Remainder $r = 0$ This zero remainder is the mathematical definition of a factor

In this scenario, our equation simplifies to:

n = 2q + 0 \implies n = 2q

(This form guarantees divisibility by two)

What numbers does this form generate? Let's plug in some whole number values for $q$ :

If $q = 1$ , then $n = 2 (1) = 2$
If $q = 2$ , then $n = 2 (2) = 4$
If $q = 3$ , then $n = 2 (3) = 6$
If $q = 4$ , then $n = 2 (4) = 8$

Key Observation: These are the even numbers.

This makes perfect sense—since $2$ times any whole number is always even, any number that can be written in the form $2q$ even formUse this form to express even numbers in proofs is automatically categorized as an even number.

✍️ Yes/No

Yes or No?

Is a number in the form

2q

divisible by

2

Case 2: When the Remainder $r = 1$ Odd integers always have this remainder when divided by two

Now, let's look at the only other possibility for a remainder when dividing by $2$ :

n = 2q + 1

(Your standard tool for defining odd numbers)

Testing this form with different values of $q$ :

If $q = 0$ , then $n = 2 (0) + 1 = 1$
If $q = 1$ , then $n = 2 (1) + 1 = 3$
If $q = 2$ , then $n = 2 (2) + 1 = 5$
If $q = 3$ , then $n = 2 (3) + 1 = 7$

Key Observation: These are the odd numbers.

Why does this happen? Every number sitting right next to an even number is odd. Since we know $2 q$ is always even, adding $1$ to it ( $2q + 1$ oddIt represents the very next number after an even one) will always result in an odd number.

✍️ FIB

Fill in the blank

Calculate

n

q = 10

in the form

n = 2q + 1

When we divide any integer by 2, there are only two possible remainders: 0 or 1. This splits all integers into two non-overlapping categories — even and odd — and the number line makes this alternating pattern visible.

✍️ FIB

Fill in the blank

When you divide

1, 3, 5, 7,

9

2

, what is the remainder?

✍️ Yes/No

Yes or No?

Can any integer be both even AND odd?

The Power of UniquenessFor any division, only one possible pair exists in EDL

Euclid's Division Lemma (EDL) guarantees a unique pairYou are guaranteed exactly one quotient and remainder $(q, r)$ for any given integers $a$ and $b$ .

Key takeaway: For any specific division, there is only one possible quotient and one possible remainder. These values are fixedOnce you pick your numbers, quotient and remainder are set in stone and will never change.

This means when you divide a number by $2$ , the remainder is constant—it's not going to be $0$ today and $1$ tomorrow!

✍️ Yes/No

Yes or No?

If we divide

25

2

, does EDL allow for more than one possible remainder?

Mutual Exclusivity of Integers

Because the remainder is uniqueThe remainder for any division is always the same, every integer falls into exactly one categoryEvery single number belongs in exactly one bucket. For a divisor of $2$ , integers are sorted into two distinct "buckets":

Even Bucket:Where numbers with remainder zero go Remainder $r = 0$ (Form: $2 q$ )
Odd Bucket:(Where numbers with remainder one go) Remainder $r = 1$ (Form: $2 q + 1$ )

Important Rule: No integer can ever be in both buckets at the same timeIt is impossible to belong to two categories. Being in both would require having two different remaindersA number cannot have two different remainders for the same division, which EDL proves is impossible.

✍️ Yes/No

Yes or No?

If an integer is even (

2q

), can it also be odd (

2q + 1

The Power of Categorization

This is not just a specific fact about even and odd numbers — it is a universal methodA systematic approach for organizing all numbers.

Key Concept: Euclid's Division Lemma (EDL)The foundational tool for organizing the number system provides a way to categorize every integer in existenceSorting all numbers so none are left out by splitting them into distinct "buckets"Distinct groups where each number belongs based on a chosen divisor.

✍️ MCQ

Choose one

If we use

b = 3

as our divisor, how many categories of remainders will we have?

Remainder Categories

The number of categories created always matches the divisor. For example:

Dividing by 3: Gives three categoriesThe divisor determines the number of buckets (Remainders: $0, 1, 2$ (The only possible remainders when dividing by three)).
Dividing by 4: Gives four categories (Remainders: $0, 1, 2, 3$ ).
Dividing by 10: Gives ten categories.

The Rule: Each number fits into exactly one bucketA number cannot have two different remainders based on its remainder, ensuring no integer is left out.

✍️ Yes/No

Yes or No?

Is the number of categories always equal to the divisor

b

4. Every integer is 3m, 3m + 1, or 3m + 2

You just proved that dividing by $2$ splits all integers into two categories:

Even
Odd

The same method works for any divisor.

Let's try it with $3$ and see what categories appear.

You know that dividing by $b$ gives exactly $b$ possible remainders, and that EDL (Euclid's Division Lemma) guarantees each integer falls into exactly one category.

✍️ Question

Your turn 🤔

Using EDL with divisor $3$ , explain why every positive integer must be in one of three forms:

$3 m$
$3 m + 1$
$3 m + 2$

Why can there not be a fourth form?

Applying EDL with Divisor $b = 3$

Having explored even and odd numbers (where the divisor was $2$ ), let's take things up a notch by applying Euclid's Division Lemma (EDL)The divisor controls how many forms we can write with a divisor $b = 3$ .

Take any positive integer $n$ . According to EDL, $n$ can be expressed as:

$n = 3 q + r$

The Remainder Constraint

Crucially, the remainder $r$ must follow this familiar constraint: $0 \leq r < 3$

This means the remainder starts at zero and must be strictly lessRemainder can never reach or exceed the divisor than the divisor.

✍️ Yes/No

Yes or No?

Can the remainder

r

be equal to

3

when the divisor is

3

The Possible Values of $r$

The constraint $0 \leq r < 3$ means $r$ is a whole number that is at least $0$ and strictly less than $3$ .

Starting at zero and counting up, the only whole numbers in that range are:

$r = 0$ One of the three possible remainders
$r = 1$ (One of the three possible remainders)
$r = 2$ One of the three possible remainders

Key Insight: There are exactly three possibilities3!Because our divisor is three, no more and no less. The number of possible remainders is always equal to the divisor $b$ Remainders always equal the divisor count.

✍️ MCQ

Choose one

Which of these is a valid remainder

r

for

b = 3

Interpreting the Remainder

When we divide an integer $n$ by $3$ using Euclid's Division Lemma ( $n = 3 q + r$ ), the remainder $r$ It tells you exactly which group a number belongs to tells us exactly how that number relates to $3$ .

Case 1: $r = 0$
- Equation: $n = 3q + 0 = 3q$ No remainder means an exact multiple of 3
- Meaning: The number is an exact multiple of 3The number divides evenly by 3.
- Logic: $n$ is exactly three times some whole number $q$ .
- Examples: $3, 6, 9, 12, \dots$

✍️ Yes/No

Yes or No?

Is the number

15

in the form

3q

Dealing with Leftovers

When there is a remainder, it represents the "distance" from the previous multiple of 3.

Case 2: $r = 1$
- Equation: $n = 3q + 1$ The number is one more than a multiple of 3
- Meaning: The number is one more than a multiple of 3One step past a multiple of 3.
- Example: $7$ is $6 + 1$ (where $6$ is the multiple).
- List: $1, 4, 7, 10, \dots$
Case 3: $r = 2$
- Equation: $n = 3q + 2$ The number is two more than a multiple of 3
- Meaning: The number is two more than a multiple of 3Two steps past a multiple of 3.
- Example: $5$ is $3 + 2$ or $8$ is $6 + 2$ .
- List: $2, 5, 8, 11, \dots$

Summary: Every single integer in existence falls into one of these three categories!No other options when dividing by 3

✍️ MCQ

Choose one

Which form does the number

8

represent?

The Pattern of Integers

Look at the number line on your screen—you can see exactly how the integers cycle through these three categories in a repeating pattern:

Multiple of 3Where the pattern resets ( $3 q$ )
One more ( $3 q + 1$ )
Two more ( $3 q + 2$ )

The Cycle: Notice the rhythmThe repeating pattern every three numbers? It goes: Multiple $\to$ One more $\to$ Two more $\to$ Restart. Every three numbers, we land back on a multiple of 3.This cycle covers every possible number This cycle ensures that no integer is left out!

Why only three forms?

You might be wondering why we only have three categories ( $3 q, 3 q + 1, 3 q + 2$ ) and not a fourth or fifth one.

The Mathematical Constraint

To have a fourth form, we would need the remainder $r$ to be $3$ . However, in Euclid's Division Lemma, the remainder must follow a strict rule:

The Rule: $0 \leq r < b$

Because the inequality is strict ( $r < 3$ ), $r$ cannot be equal to $3$ . Since $3$ is not less than $3$ The remainder can never equal or exceed the divisor, that form is mathematically impossibleA fourth category cannot exist.

✍️ Yes/No

Yes or No?

In the constraint

0 \leq r < 3

, can the remainder

r

ever be equal to

3

Physical Interpretation

Think about this practically: imagine you are sharing chocolates in groups of 3.

If you have $3$ chocolates left over on the table, you wouldn't call that a remainder.
You would simply form one more full groupThe leftover chocolates make another complete group!

Key Insight

A remainder of $3$ is actually just a remainder of $0$ in disguise(Three left over means zero actual remainder). If $r = b$ , it simply means the division was not finishedWhen remainder equals divisor, keep dividing.

✍️ MCQ

Choose one

If you are dividing by

5

and have

5

items 'left over', how many more full groups can you form?

The Power of Case-Analysis

This is how case-analysis proofs work:

Constraint Creation: Using Euclid's Division Lemma, a specific divisor (the constraint) creates a finite number of categories based on remainders.
Exhaustive Proof: To prove a statement for all numbers, you simply prove it for each of these categories one by one.

Each category is called a case. By covering every possible case, you prove the result for the entire "universe" of numbers.

✍️ MCQ

Choose one

If we use

b = 4

, how many cases must we analyze?

Real-World Application

You will use this technique to prove results like:

"Among any three consecutive integersThe statement we want to prove, one must be divisible by 3One of them must be a multiple of three."

Example: In the sequence $5,$ $6,$ $7$ , the number $6$ is a multiple of $3$ .

The Strategy: Instead of checking every number in the universe, we analyze the three categoriesTesting these three forms covers every scenario created by a divisor of $3$ :

Case 1: $3 q$
Case 2: $3 q + 1$
Case 3: $3 q + 2$

By testing these three forms, we can prove the rule once and for allMost efficient way without testing individual numbers.

✍️ MCQ

Choose one

Which of these are consecutive integers?

Factors, Multiples, and How EDL Splits Integers into Categories

1. Factors and multiples from r = 0

Key Terms:

Question 🤔

Case: Zero Remainder (r=0r = 0r=0)

The Relationship: Factors and Multiples

Understanding Factors

Definition

Example

Other Factors of 63

Important Observations

What is a Multiple?

Multiples of 7

The Shortcut to Remembering

Think of it like a recipe:

Applying the Shortcut

Equivalent Phrasings in Division

1. "Divides"

2. "Divisible By"

The Secret of the Zero Remainder

r=0r = 0r=0key(The critical condition for factors and multiples)

2. Counting possible remainders

A Key Question

Question 🤔

The Remainder Constraint

The Remainder-Divisor Relationship

Two Crucial Remainder Rules

1. r=0r = 0r=0 is included

2. rrr can never equal the divisor

The Universal Remainder Rule

The Integer "Sorting Machine"Categorizes every integer by remainder

3. Every integer is even or odd

Question 🤔

The Logic of Categories

Constraints on the Remainder

Case 1: When the Remainder r=0r = 0r=0This zero remainder is the mathematical definition of a factor

Case 2: When the Remainder r=1r = 1r=1Odd integers always have this remainder when divided by two

The Power of UniquenessFor any division, only one possible pair exists in EDL

Mutual Exclusivity of Integers

The Power of Categorization

Remainder Categories

4. Every integer is 3m, 3m + 1, or 3m + 2

Your turn 🤔

Applying EDL with Divisor b=3b = 3b=3

The Remainder Constraint

The Possible Values of rrr

Interpreting the Remainder

Dealing with Leftovers

The Pattern of Integers

Why only three forms?

The Mathematical Constraint

Physical Interpretation

Key Insight

The Power of Case-Analysis

Real-World Application

Case: Zero Remainder ( $r = 0$ )

$r = 0$
key(The critical condition for factors and multiples)

1. $r = 0$ is included

2. $r$ can never equal the divisor

Case 1: When the Remainder $r = 0$ This zero remainder is the mathematical definition of a factor

Case 2: When the Remainder $r = 1$ Odd integers always have this remainder when divided by two

Applying EDL with Divisor $b = 3$

The Possible Values of $r$