Number Theory and Fractions Lesson

Lesson Overview

• Factors and Multiples
• Prime and Composite Numbers
• Prime Factorization
• Greatest Common Factor (GCF)
• Least Common Multiple (LCM)
• Types of Fractions
• Equivalent Fractions
• Simplifying Fractions
• Finding a Fraction of a Whole Number
• Adding Like Fractions

Math is not just about counting-it's also about understanding how numbers work together. Number theory helps us break down numbers into parts, see how they connect, and recognize patterns. 

Fractions show us how to divide something into equal pieces or share parts of a whole. In this lesson, we'll explore both topics side by side.

By the end, you'll be able to:

• Find factors and multiples
  
• Identify prime and composite numbers
  
• Use greatest common factor (GCF) and least common multiple (LCM)
  
• Work confidently with proper and improper fractions
  
• Simplify, compare, and add like fractions
  
• Find a part of a whole using fractions
  

Let's begin with number theory.

Factors and Multiples

🔹 Factors

A factor of a number is a number that divides it exactly, with no remainder.

Example:
Factors of 12:

• 1 × 12 = 12
  
• 2 × 6 = 12
  
• 3 × 4 = 12
  So, the factors are: 1, 2, 3, 4, 6, 12
  

🔹 Multiples

A multiple is what you get when you multiply a number by any whole number.

Example:
Multiples of 4:

• 4 × 1 = 4
  
• 4 × 2 = 8
  
• 4 × 3 = 12
  So, the first five multiples of 4 are: 4, 8, 12, 16, 20
  

Key tip:

• Every number has at least two factors (1 and itself)
  
• Every number has infinite multiples
  

Prime and Composite Numbers

🔹 Prime Numbers

A prime number has exactly two factors: 1 and itself.

Examples:
2, 3, 5, 7, 11, 13, 17...

• 2 is the only even prime number
  
• All other even numbers are divisible by 2, so they are not prime
  

🔹 Composite Numbers

A composite number has more than two factors.

Examples:
4 (1, 2, 4),
6 (1, 2, 3, 6),
8 (1, 2, 4, 8)

🔸 Special Case: 1

• The number 1 is neither prime nor composite
  
• It has only one factor: 1
  

Learning how to identify primes and composites helps in breaking numbers into their prime factors.

Prime Factorization

Prime factorization means writing a number as a product of prime numbers only.

Use a factor tree to break a number into its prime parts.

Example: Prime factorization of 36

36 = 6 × 6
6 = 2 × 3
So, 36 = 2 × 3 × 2 × 3
Group and write: 2² × 3²

You can double-check by multiplying back:
2 × 2 = 4
3 × 3 = 9
4 × 9 = 36

Prime factorization is used in finding GCF and LCM.

Take This Quiz:

Take The Prime Factorization Math Test!

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest number that is a factor of two or more numbers.

Example: Find the GCF of 18 and 24

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6
GCF = 6

You can also find the GCF using prime factorization:

• 18 = 2 × 3 × 3
  
• 24 = 2 × 2 × 2 × 3
  Common primes: 2 × 3 = 6
  

GCF is useful in simplifying fractions.

Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest multiple shared by two or more numbers.

Example: LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18, 24...
Smallest common multiple = 12

Another method:

• Use prime factorization
  
• Multiply all prime factors, using the highest powers for each
  

LCM is important when adding or subtracting fractions with different denominators.

Take This Quiz:

Number Theory and Fractions

Types of Fractions

🔸 Proper Fractions

The numerator is less than the denominator.
Example: 3/4

🔸 Improper Fractions

The numerator is greater than or equal to the denominator.
Examples: 5/3, 7/4

🔸 Mixed Numbers

A whole number and a fraction together.
Example: 2 1/2

To change an improper fraction to a mixed number:

• Divide the numerator by the denominator
  
• Write the remainder as a fraction
  

Example: 9/4

9 ÷ 4 = 2 remainder 1 → 2 1/4

Understanding types of fractions helps you identify and convert them easily.

Equivalent Fractions

Equivalent fractions are different-looking fractions that show the same amount.

To find an equivalent fraction:

• Multiply or divide the numerator and denominator by the same number
  

Example:

1/2 = (1 × 2)/(2 × 2) = 2/4
Also: 3/5 = 6/10 = 9/15

To check if two fractions are equivalent, cross-multiply:

• 1/2 and 2/4
  1 × 4 = 4
  2 × 2 = 4 → Both equal → Equivalent
  

Equivalent fractions are useful in comparing and simplifying.

Simplifying Fractions

Simplifying means writing the smallest possible version of a fraction that keeps its value the same.

Example:

12/16
GCF of 12 and 16 is 4
Divide both by 4:
12 ÷ 4 = 3, 16 ÷ 4 = 4
Simplified fraction = 3/4

Always try to simplify your answer when solving a fraction problem.

Finding a Fraction of a Whole Number

To find a fraction of a number, divide first, then multiply.

Formula:

Fraction of a number = (Whole number ÷ Denominator) × Numerator

Example:

What is 3/4 of 20?
20 ÷ 4 = 5
5 × 3 = 15

Another example: Find 2/5 of 30
30 ÷ 5 = 6
6 × 2 = 12

This is a very useful skill for solving word problems.

Take This Quiz:

Fractions - Quiz

Adding Like Fractions

When the denominator is the same, just:

• Add the numerators
  
• Keep the denominator
  

Example:

2/5 + 1/5 = (2 + 1)/5 = 3/5

If the sum is improper, convert it to a mixed number.

Example:

5/6 + 4/6 = 9/6
9 ÷ 6 = 1 remainder 3 → 1 3/6 → Simplified: 1 1/2

Steps:

1. Add numerators
   
2. Keep denominator
   
3. Simplify or convert if needed
   

This is the first step before learning how to add unlike fractions.

Take This Quiz:

Fraction Math Quiz