GCF and LCM

🔢 GCF and LCM

Part 1 of 5 — Factors and the Greatest Common Factor

Topics in This Part

🔑 Key Concept: A factor of a number divides into it evenly, with no remainder. The Greatest Common Factor (GCF) is the biggest factor that two numbers share. Everything in this part is built from one idea: which numbers divide evenly?

What Is a Factor?

A factor of a number is any whole number that divides into it evenly — leaving a remainder of $0$.

$12 \div 3 = 4 \text{ (no remainder)} \;\Rightarrow\; 3 \text{ is a factor of } 12$

Factors always come in pairs that multiply to the number. For $12$:

So the factors of $12$ are: $1,\ 2,\ 3,\ 4,\ 6,\ 12$.

💡 Tip: Every number has $1$ and itself as factors. To list them all, walk up from $1$ and write down each pair as you find it.

Concept Check 🎯

A Tidy Way to List Every Factor

To be sure you catch all the factors, test the whole numbers in order — $1, 2, 3, \dots$ — and write down each pair you find. Stop when the pair starts to repeat.

Example: factors of $20$

So the factors of $20$ are $1, 2, 4, 5, 10, 20$.

💡 Once the two numbers in a pair "cross over" (here $5 \times 4$ repeats $4 \times 5$), you have found them all.

List the Factors 🧮

Find the factors by checking pairs that multiply to the number.

1) How many factors does $10$ have in total? (list: $1, 2, 5, 10$) 2) What is the largest factor of $20$ that is less than $20$ itself? 3) Is $7$ a factor of $21$? Enter 1 for yes or 0 for no.

Common Factors and the GCF

A common factor is a number that is a factor of two numbers at once. The Greatest Common Factor (GCF) is the largest one they share.

Example: GCF of $12$ and $18$

The factors they share are $1, 2, 3, 6$. The greatest of these is $6$.

$\text{GCF}(12, 18) = 6$

🔑 List-and-Compare Method: List every factor of each number, circle the ones that appear in both lists, and pick the biggest. That biggest shared factor is the GCF.

Find the GCF by Listing 🧮

List the factors of each number, then find the greatest factor they share.

1) GCF of $8$ and $12$ $= \,?$ 2) GCF of $9$ and $15$ $= \,?$ 3) GCF of $6$ and $10$ $= \,?$

🔢 GCF and LCM

Part 2 of 5 — Prime Factorization & a Faster GCF

🔑 The Idea: Listing every factor is slow for big numbers. Instead, break each number into its prime building blocks — its prime factorization — and find the GCF from the parts they share.

Prime Numbers and Factor Trees

A prime number has exactly two factors: $1$ and itself. The first few primes are:

$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ \dots$

A number that is not prime (and isn't $1$) is composite — it can be broken down further.

Prime factorization rewrites a number as a product of primes only. Build it with a factor tree:

$24 = 4 \times 6 = (2 \times 2) \times (2 \times 3) = 2 \times 2 \times 2 \times 3$

Using exponents: $24 = 2^3 \times 3$.

💡 Check: $2 \times 2 \times 2 \times 3 = 8 \times 3 = 24$ ✓. No matter how you split a number, the prime factorization always comes out the same.

Concept Check 🎯

Building a Factor Tree

Start by splitting the number into any two factors, then keep splitting until every branch ends in a prime.

$30 \to 5 \times 6 \to 5 \times (2 \times 3) = 2 \times 3 \times 5$

It doesn't matter where you start — $30 = 3 \times 10 = 3 \times (2 \times 5)$ gives the same primes $2 \times 3 \times 5$.

🔑 Goal: Keep breaking down composite numbers until only primes are left on the branches. Those primes, multiplied together, are the prime factorization.

Build the Prime Factorization 🧮

Break each number down to primes. Enter the missing value.

1) $18 = 2 \times 3 \times \,?$ 2) $20 = 2 \times 2 \times \,?$ 3) $30 = 2 \times 3 \times \,?$

Finding the GCF from Prime Factors

To find the GCF, line up the prime factorizations and multiply the primes they share (taking the smaller power of each shared prime).

Example: GCF of $24$ and $36$

$24 = 2 \times 2 \times 2 \times 3 \qquad 36 = 2 \times 2 \times 3 \times 3$

Multiply the shared parts: $2 \times 2 \times 3 = 12$.

$\text{GCF}(24, 36) = 12$

⚠️ Watch out: Only multiply primes that appear in both numbers, and only as many times as the number with fewer of them. The extra $2$ in $24$ and the extra $3$ in $36$ are not shared, so they're left out.

Find the GCF of $40$ and $60$ 🔽

Use the prime factorizations $40 = 2 \times 2 \times 2 \times 5$ and $60 = 2 \times 2 \times 3 \times 5$.

The GCF Recipe (Three Steps)

🔑 To find any GCF:

1. Write each number's prime factorization.
2. Circle the primes that appear in both, taking the smaller count of each.
3. Multiply the circled primes.

If two numbers share no primes, the only common factor is $1$ — so their GCF is $\mathbf{1}$. (We call such numbers relatively prime: for example, $8 = 2^3$ and $9 = 3^2$ share nothing, so $\text{GCF}(8, 9) = 1$.) Now try the recipe yourself.

GCF with Prime Factorization 🧮

Find each GCF using the prime building blocks.

1) GCF of $16$ and $24$ $= \,?$ 2) GCF of $18$ and $30$ $= \,?$ 3) GCF of $14$ and $35$ $= \,?$

🔢 GCF and LCM

Part 3 of 5 — Multiples and the Least Common Multiple

🔑 The Switch: GCF was about dividing — the biggest number that fits inside both. The LCM flips it: the smallest number that both numbers fit inside. It's the smallest number on both of their times tables.

What Is a Multiple?

A multiple of a number is what you get when you multiply it by $1, 2, 3, 4, \dots$ — its times table.

A common multiple appears in both lists. For $4$ and $6$, both lists contain $12$ and $24$. The Least Common Multiple (LCM) is the smallest common multiple.

$\text{LCM}(4, 6) = 12$

💡 Factor vs. Multiple: A factor is smaller than (or equal to) the number; a multiple is larger than (or equal to) the number. Factors divide in; multiples count up.

Concept Check 🎯

Listing Multiples to Find the LCM

The simplest way to find an LCM with small numbers: list the multiples of each number and stop at the first value that shows up in both lists.

Example: LCM of $6$ and $8$

The first match is $24$, so $\text{LCM}(6, 8) = 24$.

💡 Shortcut: If the larger number is already a multiple of the smaller (like $9$ and $3$), the larger number is the LCM — no listing needed.

Find the LCM by Listing 🧮

List multiples of each number until you hit the first one they share.

1) LCM of $2$ and $5$ $= \,?$ 2) LCM of $4$ and $6$ $= \,?$ 3) LCM of $3$ and $9$ $= \,?$

Finding the LCM from Prime Factors

For bigger numbers, use prime factorizations. For the LCM, take each prime the greatest number of times it appears in either number — then multiply.

Example: LCM of $12$ and $18$

$12 = 2 \times 2 \times 3 \qquad 18 = 2 \times 3 \times 3$

Multiply: $2 \times 2 \times 3 \times 3 = 36$.

$\text{LCM}(12, 18) = 36$

⚠️ GCF vs. LCM — opposite rules! For the GCF you take the smaller count of each shared prime. For the LCM you take the larger count of every prime that appears. Don't mix them up.

Find the LCM of $8$ and $12$ 🔽

Use $8 = 2 \times 2 \times 2$ and $12 = 2 \times 2 \times 3$.

The LCM Recipe (Three Steps)

🔑 To find any LCM:

1. Write each number's prime factorization.
2. For every prime that appears, take the larger count.
3. Multiply them all together.

If two numbers share no primes, just multiply them — for example $\text{LCM}(5, 6) = 5 \times 6 = 30$. Your turn.

LCM Practice 🧮

Find each LCM (listing or prime factors — your choice).

1) LCM of $8$ and $10$ $= \,?$ 2) LCM of $5$ and $6$ $= \,?$ 3) LCM of $10$ and $15$ $= \,?$

🔢 GCF and LCM

Part 4 of 5 — Word Problems: GCF or LCM?

🔑 The Hard Part Isn't the Math — It's Choosing. Most mistakes happen when students compute the GCF when the problem actually wanted the LCM (or vice versa). This part teaches you how to tell them apart.

How to Decide: GCF or LCM?

Read the problem and ask what's happening:

🔑 Quick Test:

• "What is the greatest/largest/biggest number that…?" → almost always GCF.
• "When will they happen together again?" or "smallest/least number that…?" → almost always LCM.

Which Tool? 🎯

Worked Example — A GCF Problem

Problem: You have $18$ red beads and $24$ blue beads. You want to make identical bracelets that use up all the beads, with the greatest number of bracelets. How many bracelets, and what's on each?

Step 1 — Choose the tool. "Greatest number of equal groups, no leftovers" → GCF.

Step 2 — Find the GCF. $18 = 2 \times 3 \times 3 \qquad 24 = 2 \times 2 \times 2 \times 3$ Shared primes: one $2$ and one $3$ → $\text{GCF} = 2 \times 3 = 6$.

Step 3 — Answer. You can make $6$ bracelets. Each gets: $18 \div 6 = 3 \text{ red beads}, \qquad 24 \div 6 = 4 \text{ blue beads}.$

✅ Check: $6$ bracelets $\times 3$ red $= 18$ ✓ and $6 \times 4 = 24$ ✓ — all beads used, none left over.

Worked Example — An LCM Problem

Problem: Hot dogs come in packs of $10$; buns come in packs of $8$. What is the smallest number of each you can buy so there are no leftovers of either?

Step 1 — Choose the tool. "Smallest number where both totals match" → LCM.

Step 2 — Find the LCM. $10 = 2 \times 5 \qquad 8 = 2 \times 2 \times 2$ Take the most of each prime: three $2$s and one $5$ → $\text{LCM} = 2 \times 2 \times 2 \times 5 = 40$.

Step 3 — Answer. Buy $40$ hot dogs and $40$ buns. That's: $40 \div 10 = 4 \text{ packs of hot dogs}, \qquad 40 \div 8 = 5 \text{ packs of buns}.$

✅ Check: $4 \times 10 = 40$ hot dogs and $5 \times 8 = 40$ buns — they match exactly, with none left over.

Solve a Word Problem 🔽

Two lighthouses flash on a cycle: one every $6$ seconds, the other every $9$ seconds. They just flashed together.

Keyword Signals

When a problem is wordy, hunt for these signals before you compute:

⚠️ Don't be fooled by the big numbers. A GCF answer is usually smaller than the numbers in the problem; an LCM answer is usually bigger. If your answer feels like it's going the wrong direction, re-read which tool the problem needs.

Word Problem Drill 🧮

Decide GCF or LCM, then solve.

1) A teacher splits $28$ pencils and $42$ erasers into identical kits with none left over, using the most kits possible. How many kits? 2) A red light blinks every $12$ s and a green light every $16$ s. After how many seconds do they blink together again?

🔢 GCF and LCM

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) list factors and multiples, (2) build prime factorizations, (3) find the GCF and the LCM two ways, and (4) decide which one a word problem needs. Let's put it all together.

Quick Reference

💡 Handy fact: For any two numbers, $\text{GCF} \times \text{LCM} =$ the product of the two numbers. Check with $12$ and $18$: $\text{GCF} = 6$, $\text{LCM} = 36$, and $6 \times 36 = 216 = 12 \times 18$ ✓.

⚠️ The single most common error is picking the wrong tool. GCF makes things smaller; LCM makes things bigger. Re-read the question and ask which way the answer should go.

Mixed Practice 🎯

One Quick Habit Before You Compute

In the next drill, each item says either GCF or LCM right in the prompt — but read the whole prompt anyway. On a real test the word might be hidden in a story.

🔑 Two-second self-check: Before circling an answer, ask: "Did the problem want the answer to come out smaller (GCF) or bigger (LCM)?" That one question prevents the most common mistake.

Mixed Drill 🧮

1) GCF of $16$ and $40$ $= \,?$ 2) LCM of $9$ and $12$ $= \,?$ 3) GCF of $21$ and $28$ $= \,?$

You're Ready

That's the whole toolkit: factors and multiples, prime factorization, the GCF recipe, the LCM recipe, and the word-problem decision. The exit quiz below pulls from all five parts — take your time and use your self-check.

💡 Final reminder: GCF → multiply shared primes (smaller counts) → answer comes out small. LCM → multiply all primes (larger counts) → answer comes out big.

Exit Quiz ✅

Answer all three to finish the lesson.