Factors and Multiples

🧩 Factors and Multiples

Part 1 of 5 — Finding Factors

Topics in This Part

🔑 Key Concept: A factor of a number divides into it evenly — with no remainder left over. Learning to list all of a number's factors is the first tool you'll use in this whole lesson.

What Is a Factor?

A factor of a number is a whole number that divides into it with no remainder.

Think about $12$. Which numbers divide into it evenly?

$12 \div 1 = 12 \quad 12 \div 2 = 6 \quad 12 \div 3 = 4 \quad 12 \div 4 = 3 \quad 12 \div 6 = 2 \quad 12 \div 12 = 1$

Every one of those came out to a whole number, so the factors of $12$ are:

$1,\; 2,\; 3,\; 4,\; 6,\; 12$

What about $12 \div 5 = 2\text{ R }2$? That leaves a remainder, so $5$ is not a factor of $12$.

🔑 Two facts that are always true: $1$ is a factor of every number, and every number is a factor of itself. So the smallest factor is always $1$ and the largest is always the number itself.

Concept Check 🎯

Factor Pairs — Find Them Two at a Time

Factors come in pairs that multiply to the number. To find every factor of $24$, start at $1$ and work up, writing each pair:

Once the pairs start to repeat (the next would be $6 \times 4$, which we already have), you're done. Reading the columns gives all the factors of $24$:

$1,\; 2,\; 3,\; 4,\; 6,\; 8,\; 12,\; 24$

💡 Why pairs help: you only have to test the small numbers. Each small factor hands you its big partner for free, so you never miss one and never double-count.

Complete the Factor Pair 🧮

Each pair multiplies to the target number. Find the missing partner.

1) $18 = 2 \times \,?$ 2) $18 = 3 \times \,?$ 3) $36 = 4 \times \,?$

Reading the Whole Factor List

Once you've listed every factor pair, reading down the columns gives the complete list of factors — already in order from smallest to largest.

For $20$: the pairs are $1\times20$, $2\times10$, $4\times5$. So the factors are $1, 2, 4, 5, 10, 20$ — that's six factors.

💡 One careful spot: when a number is a "square" like $16 = 4\times4$, the repeated factor ($4$) is written only once. That's why $16$ has an odd number of factors.

Count the Factors 🧮

List the factors in your head using factor pairs, then enter how many factors each number has.

1) How many factors does $16$ have? (Hint: $1\times16,\ 2\times8,\ 4\times4$) 2) How many factors does $15$ have?

🧩 Factors and Multiples

Part 2 of 5 — Finding Multiples

🔑 The Idea: A multiple of a number is what you get when you multiply it by $1, 2, 3, 4, \ldots$ Multiples are the "skip-counting" numbers — and there are infinitely many of them.

What Is a Multiple?

A multiple of a number is the product of that number and any whole number ($1, 2, 3, \ldots$).

The multiples of $4$ are just the times-$4$ table:

$4 \times 1 = 4 \quad 4 \times 2 = 8 \quad 4 \times 3 = 12 \quad 4 \times 4 = 16 \quad 4 \times 5 = 20$

So the first five multiples of $4$ are:

$4,\; 8,\; 12,\; 16,\; 20,\; \ldots$

You can also find them by skip-counting: start at $4$ and keep adding $4$.

🔑 Key fact: The list of multiples never ends — you can always multiply by a bigger number. The smallest multiple of any number is the number itself (multiply by $1$).

List the Multiples 🧮

Skip-count to fill in the missing multiples.

1) Multiples of $6$: $6,\ 12,\ \underline{\;?\;},\ 24,\ 30$ 2) Multiples of $7$: $7,\ 14,\ 21,\ \underline{\;?\;},\ 35$ 3) The $4$th multiple of $9$ is $\underline{\;?\;}$

Factor vs. Multiple — Don't Mix Them Up!

These two words are easy to confuse. Use $3$ and $12$ to see the difference:

Notice they describe the same relationship from two directions:

$\underbrace{3}_{\text{factor (small)}} \longrightarrow \underbrace{12}_{\text{multiple (big)}}$

💡 Memory trick: A factor fits inside a number (it's small). A multiple is many of a number (it's big — or equal). Factors are at or below the number; multiples are at or above it.

Factor or Multiple? 🔽

Choose the word that makes each sentence true.

How to Spot a Multiple

To check whether a big number is a multiple of a small one, just divide. If it comes out even (no remainder), it's a multiple.

Is $32$ a multiple of $8$? $\;32 \div 8 = 4$ ✓ — yes. Is $28$ a multiple of $8$? $\;28 \div 8 = 3\text{ R }4$ ✗ — no.

🔑 The deep connection: "$32$ is a multiple of $8$" and "$8$ is a factor of $32$" are the same true fact. Dividing tests both at once.

Concept Check 🎯

🧩 Factors and Multiples

Part 3 of 5 — Prime, Composite & Prime Factorization

🔑 Why it matters: Sorting numbers by how many factors they have splits every whole number into prime or composite. Then we break composite numbers into their prime building blocks — a skill you'll lean on for GCF and LCM.

Prime vs. Composite

We can sort numbers by how many factors they have:

• A prime number has exactly two factors: $1$ and itself.
• A composite number has more than two factors.
• The number $1$ is neither prime nor composite — it has only one factor (itself).

The first few primes are $2, 3, 5, 7, 11, 13, 17, 19, \ldots$

⚠️ Watch out: $2$ is the only even prime — every other even number is divisible by $2$, giving it a third factor. And $1$ is not prime, even though it feels like it should be.

Sort Each Number 🔽

Decide whether each number is prime, composite, or neither.

Prime Factorization with a Factor Tree

Every composite number can be written as a product of primes only. A factor tree breaks it down step by step. Let's factor $36$:

$36 = 4 \times 9$ $4 = 2 \times 2 \qquad 9 = 3 \times 3$

Keep splitting until every branch ends in a prime. The primes at the bottom are:

$36 = 2 \times 2 \times 3 \times 3$

We can write repeated primes with exponents: $\;36 = 2^2 \times 3^2$.

💡 It doesn't matter how you start! You could begin $36 = 6 \times 6$ instead, and you'd still end with $2 \times 2 \times 3 \times 3$. Every number has exactly one prime factorization.

Worked Example: Prime Factorization of $24$

Split off the smallest prime each time:

$24 = 2 \times 12$ $12 = 2 \times 6$ $6 = 2 \times 3$

All branches now end in primes, so:

$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$

✅ Check: $2 \times 2 \times 2 \times 3 = 8 \times 3 = 24$ ✓

Concept Check 🎯

Filling In a Missing Prime

If you already know most of a number's prime factorization, you can find a missing prime by dividing out the primes you have.

Suppose $28 = 2 \times 2 \times \,?$. Multiply the primes you know: $2 \times 2 = 4$. Then $28 \div 4 = 7$, so the missing prime is $7$:

$28 = 2 \times 2 \times 7$

✅ Check: $2 \times 2 \times 7 = 4 \times 7 = 28$ ✓

Build the Factorization 🧮

Finish each prime factorization by filling in the missing prime.

1) $12 = 2 \times 2 \times \,?$ 2) $20 = 2 \times 2 \times \,?$ 3) $45 = 3 \times 3 \times \,?$

🧩 Factors and Multiples

Part 4 of 5 — Greatest Common Factor (GCF)

🔑 Big Idea: The Greatest Common Factor of two numbers is the largest number that is a factor of both. It's the key to simplifying fractions and sharing things into equal groups.

Finding the GCF by Listing Factors

To find the GCF of two numbers, list the factors of each, circle the ones they share, and pick the biggest.

Example: GCF of $12$ and $18$

The common factors (in both lists) are $1, 2, 3, 6$. The greatest of these is:

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

🔑 What "common" means: a common factor divides both numbers. The greatest common factor is simply the largest one they share.

Concept Check 🎯

A Faster Way: Use Prime Factorizations

For bigger numbers, listing every factor is slow. Instead, write each number's prime factorization and multiply the primes they share.

Example: GCF of $24$ and $36$

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

Match up the primes they have in common: two $2$'s and one $3$.

$\text{GCF} = 2 \times 2 \times 3 = 12$

💡 The rule: for each prime, take the smaller count it appears in the two numbers. $24$ has three $2$'s and $36$ has two $2$'s, so you take two $2$'s. Both have one $3$, so take one $3$.

Find the GCF 🧮

List the factors (or use prime factorizations) and enter the greatest common factor.

1) GCF of $16$ and $24$ $= \,?$ 2) GCF of $20$ and $30$ $= \,?$ 3) GCF of $14$ and $21$ $= \,?$

A Handy Shortcut

When the smaller number is already a factor of the larger one, the GCF is just the smaller number itself.

For example, $6$ and $18$: since $18 \div 6 = 3$ evenly, $6$ divides both. The largest factor of $6$ is $6$, so:

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

💡 Sanity check on any GCF: it can never be larger than the smaller of the two numbers. If you ever get a "GCF" bigger than both, you've made a slip.

Walk Through a GCF 🔽

Find the GCF of $18$ and $24$ step by step.

🧩 Factors and Multiples

Part 5 of 5 — Least Common Multiple (LCM) & Mastery Check

🔑 Last Tool: The Least Common Multiple of two numbers is the smallest number that is a multiple of both. It's exactly what you need to add fractions and to solve "when do two cycles line up?" problems.

Finding the LCM by Listing Multiples

To find the LCM, list multiples of each number and grab the first one that appears in both lists.

Example: LCM of $4$ and $6$

The first multiple in both lists is $12$:

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

⚠️ Don't confuse LCM with GCF! The GCF is a factor (small — it divides the numbers). The LCM is a multiple (big — the numbers divide into it). For $4$ and $6$: GCF $= 2$, but LCM $= 12$.

Concept Check 🎯

A Shortcut for LCM

When the bigger number is already a multiple of the smaller one, the LCM is just the bigger number itself.

For example, $5$ and $10$: since $10$ is already a multiple of $5$, the smallest shared multiple is $10$:

$\text{LCM}(5, 10) = 10$

💡 Sanity check on any LCM: it can never be smaller than the larger of the two numbers, and it's often the two numbers multiplied together (or a bit less if they share factors).

Find the LCM 🧮

List the multiples of each number and enter the least common multiple.

1) LCM of $4$ and $10$ $= \,?$ 2) LCM of $6$ and $9$ $= \,?$ 3) LCM of $5$ and $6$ $= \,?$

Quick Reference

💡 The one-line summary: Factors are the ingredients that build a number; multiples are the numbers you build with it. GCF looks at shared ingredients; LCM looks at the first shared build.

Mixed Practice 🔽

One question from each part of the lesson.

You've Got the Whole Toolkit

You can now find factors, list multiples, sort numbers into prime and composite, break them apart with prime factorization, and find the GCF and LCM of any pair.

🔑 The big picture: factors and multiples are two views of the same multiplication fact. Master them and you've laid the foundation for simplifying fractions, finding common denominators, and much of the math ahead. One last quiz to lock it in.

Exit Quiz ✅

Answer all three to finish the lesson.