Factors, Multiples & Divisibility | Fifth Grade

Complete Notes & Formulas

1. Identify Factors

Definition: A factor of a number is a whole number that divides it exactly without leaving any remainder.

🔑 Key Concept:

If A ÷ B = C (with no remainder), then B is a factor of A

• Every number has at least two factors: 1 and itself

• Factors always come in pairs

📝 How to Find Factors:

Start with 1 (1 is a factor of every number)
Check each number up to the given number
If it divides evenly (no remainder), it's a factor
List all factors from smallest to largest

✏️ Example: Find all factors of 12

12 ÷ 1 = 12 ✓ (1 and 12 are factors)

12 ÷ 2 = 6 ✓ (2 and 6 are factors)

12 ÷ 3 = 4 ✓ (3 and 4 are factors)

12 ÷ 4 = 3 ✓ (already found)

Factors of 12: 1, 2, 3, 4, 6, 12

2. Identify Multiples

Definition: A multiple of a number is the product of that number and any whole number.

🔑 Key Concept:

Multiples of n = n × 1, n × 2, n × 3, n × 4, ...

• Every number has infinite multiples

• Every number is a multiple of itself

✏️ Examples:

Multiples of 5:

5, 10, 15, 20, 25, 30, 35, 40, ...

Multiples of 8:

8, 16, 24, 32, 40, 48, 56, 64, ...

🔑 Factors vs. Multiples:

Factors	Multiples
Divide a number	Are products of a number
Limited in number	Infinite in number
Equal to or less than number	Equal to or greater than number

3. Prime and Composite Numbers

Definition: Numbers are classified as prime or composite based on how many factors they have.

📐 Definitions:

Prime Number:

A number with EXACTLY TWO factors: 1 and itself

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...

Composite Number:

A number with MORE THAN TWO factors

Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...

💡 Special Cases:

0 and 1 are neither prime nor composite
2 is the only EVEN prime number
2 is the smallest prime number
All prime numbers (except 2) are ODD

📊 Prime Numbers 1-50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

4. Prime Factorization

Definition: Prime factorization is expressing a number as a product of its prime factors.

🔑 Two Methods:

Method 1: Division Method (Ladder Method)

Divide by the smallest prime number (start with 2)
Divide the quotient by the smallest prime possible
Continue until quotient is 1
Multiply all prime divisors together

Method 2: Factor Tree Method

Break number into any two factors
Continue breaking composite factors
Stop when all factors are prime
Multiply all prime factors together

✏️ Example: Prime Factorization of 36

Division Method:

2 | 36

2 | 18

3 | 9

3 | 3

| 1

36 = 2 × 2 × 3 × 3 = 2² × 3²

Factor Tree Method:

36 = 6 × 6

6 = 2 × 3 and 6 = 2 × 3

36 = 2 × 2 × 3 × 3 = 2² × 3²

5. Divisibility Rules

Definition: Divisibility rules are shortcuts to determine if a number can be divided by another number without doing the actual division.

📐 Complete Divisibility Rules Chart:

Divisible by	Rule	Example
2	Last digit is even (0, 2, 4, 6, 8)	24, 58, 100
3	Sum of all digits is divisible by 3	27 (2+7=9÷3)
4	Last 2 digits divisible by 4	316 (16÷4=4)
5	Last digit is 0 or 5	35, 80, 125
6	Divisible by BOTH 2 AND 3	42, 66, 108
8	Last 3 digits divisible by 8	1,024 (024÷8=3)
9	Sum of all digits is divisible by 9	81 (8+1=9÷9)
10	Last digit is 0	50, 120, 1000

6. Divisibility Rules: Word Problems

Definition: Apply divisibility rules to solve real-world problems involving sharing, grouping, and organizing.

✏️ Examples:

Problem 1: A teacher has 84 pencils. Can they be divided equally among 6 students?

Solution:

Check divisibility by 6: Must be divisible by 2 AND 3

• 84 is even ✓ (divisible by 2)

• 8 + 4 = 12, 12 ÷ 3 = 4 ✓ (divisible by 3)

Answer: Yes! 84 ÷ 6 = 14 pencils each

Problem 2: Is 345 divisible by 5?

Solution:

Last digit is 5 ✓

Answer: Yes, 345 is divisible by 5

Problem 3: A baker has 126 cookies. Can they be packed equally into boxes of 9?

Solution:

Check divisibility by 9: Sum of digits must be divisible by 9

1 + 2 + 6 = 9 ✓

Answer: Yes! 126 ÷ 9 = 14 boxes

7. Find All the Factor Pairs of a Number

Definition: A factor pair is a set of two numbers that, when multiplied together, give the original number.

🔑 Key Concept:

If A × B = C, then (A, B) is a factor pair of C

📝 Steps to Find All Factor Pairs:

Start with 1 × the number
Try 2, 3, 4... as first factor
Divide the number by each factor to find the pair
Stop when factors start repeating

✏️ Example: Factor Pairs of 24

1 × 24 = 24 → (1, 24)

2 × 12 = 24 → (2, 12)

3 × 8 = 24 → (3, 8)

4 × 6 = 24 → (4, 6)

Factor Pairs: (1, 24), (2, 12), (3, 8), (4, 6)

8. Least Common Multiple (LCM)

Definition: The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.

🔑 Three Methods to Find LCM:

Method 1: Listing Multiples Method

List multiples of each number
Find common multiples
Choose the smallest common multiple

Method 2: Prime Factorization Method

Find prime factors of each number
Take the highest power of each prime factor
Multiply all highest powers together

Method 3: Division Method (Ladder Method)

Divide by smallest prime that divides at least one number
Continue until all quotients are 1
Multiply all divisors together

✏️ Example: Find LCM of 4 and 6

Method 1: Listing Multiples:

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Multiples of 6: 6, 12, 18, 24, 30, ...

Common multiples: 12, 24, ...

LCM = 12

Method 2: Prime Factorization:

4 = 2 × 2 = 2²

6 = 2 × 3

Highest powers: 2² and 3¹

LCM = 2² × 3 = 4 × 3 = 12

LCM = 12

Method 3: Division Method:

2 | 4, 6

2 | 2, 3

3 | 1, 3

| 1, 1

LCM = 2 × 2 × 3 = 12

LCM = 12

Quick Reference Chart

Concept	Key Formula/Rule
Factor	Number that divides exactly (no remainder)
Multiple	Product of a number and any whole number
Prime Number	Has exactly 2 factors (1 and itself)
Composite Number	Has more than 2 factors
Prime Factorization	Express number as product of prime factors
Factor Pairs	Two numbers that multiply to give the number
LCM	Smallest common multiple of two or more numbers