GCF, LCM, and the Distributive Property

Lesson 1 of 1: The Number System

In this lesson:

• Find the GCF of two whole numbers (up to 100)
• Find the LCM of two whole numbers (up to 12)
• Rewrite sums using the GCF and distributive property

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Find the GCF of two whole numbers up to 100
2. Find the LCM of two whole numbers up to 12
3. Rewrite a sum using the GCF and the distributive property

Equal Baskets: What Is the GCF?

Problem: 36 apples, 48 oranges. Fill identical baskets — same contents, no leftovers. Largest number of baskets?

• You need a number that divides both 36 and 48 evenly
• The largest such number is the Greatest Common Factor

What is the biggest equal-sized group that fits both?

The Greatest Common Factor Defined

The GCF of two whole numbers is the largest number that divides both evenly.

• List all factors of each number
• Identify the common factors (factors in both lists)
• Select the greatest one

Key check: GCF is always ≤ the smaller number.

GCF by Listing Factors: Example

Find

• Common factors: 1, 2, 3, 4, 6, 12

GCF by Prime Factorization: Example

Find

Step 1: Write each as a product of primes:

Step 2: Shared primes at minimum power: and

Step 3:

GCF Special Cases Worth Knowing

• — 1's only factor is 1
• — a number divides itself
• — smaller divides larger → smaller is GCF

If GCF = 1, the numbers share no factor except 1.

You Try: Finding the GCF

Find the GCF of each pair. Show your factor lists.

List all factor pairs, circle the common factors, pick the greatest.

Pause and work these out before advancing.

GCF vs. LCM: Opposite Directions

GCF: largest number that fits into both numbers
→ Factors are ≤ the numbers

LCM: smallest number that both numbers fit into
→ Multiples are ≥ the numbers

Two Buses: What Is the LCM?

Problem: Bus A every 4 min. Bus B every 6 min. When do they next leave together?

Multiples of 4: 4, 8, 12, ...
Multiples of 6: 6, 12, ...

First match: 12 — both buses coincide at 12 min after noon.

Least Common Multiple = 12.

The Least Common Multiple Defined

The LCM of two numbers is the smallest positive multiple of both.

• List multiples of each number
• Find the first (smallest) common multiple

Key check: LCM is always ≥ the larger number.

If your answer is smaller than either number, something went wrong.

LCM on a Number Line: Visual

— first position marked by both patterns

First overlap at 12 →

LCM by Listing: Two Cases

Common factor — :

• 8: 8, 16, 24 | 12: 12, 24 → LCM = 24

Coprime (GCF = 1) — :

• 3: 3, 6, 9, 12, 15, 18, 21 | 7: 7, 14, 21 → LCM = 21

Keep listing until you find a match.

You Try: Finding the LCM

Find the LCM of each pair by listing multiples.

List several multiples of the smaller number first.

Pause and work these out before advancing.

Factoring: The Distributive Property Reversed

You know the distributive property forward:

Today we go backward — starting from the sum:

This is called factoring — pulling the shared factor out of both terms.

Rewriting : Step by Step

Step 1: : common factors 1, 2, 4 → GCF = 4

Step 2: Rewrite each term:

Step 3: Factor:

Verify: ✓

Why the GCF? Partial Factoring Fails

still shares factor 2 — factor again:

After factoring: Do terms inside share any factor? If yes — use the GCF from the start.

Factoring Practice with Two More Examples

• : → ; verify ✓
• : → ; verify ✓

Check: do 3+2 and 4+3 share any common factor? No — fully factored.

You Try: Rewriting Sums with the GCF

Rewrite each sum in factored form. Verify your answer.

Steps: (1) Find GCF. (2) Rewrite terms. (3) Factor. (4) Verify.

Pause and work these out before advancing.

Key Takeaways and Misconception Warnings

✓ GCF: largest shared factor; always ≤ smaller number

✓ LCM: smallest shared multiple; always ≥ larger number

✓ Factor: ; verify by multiplying back

GCF ≤ both; LCM ≥ both — wrong direction = wrong concept

Factor out the GCF, not just any common factor

Keep listing multiples until you find the first match

Up Next: Factoring Variable Expressions

Today: GCF and factoring with whole numbers

Next (6.EE.A.3): The same procedure applies to variable expressions

— you already know how to find this.

The arithmetic you practiced today is the foundation for algebraic factoring.