Factors and Factor Pairs

🧩 Factors and Factor Pairs

Part 1 of 5 — What Is a Factor?

Topics in This Part

🔑 Key Concept: A factor is a whole number that divides into another whole number evenly — with nothing left over. Factors are the building blocks that multiply together to make a number.

What "Factor" Means

When two whole numbers multiply to make a third number, those two numbers are factors of the result.

$3 \times 4 = 12$

Here, $3$ and $4$ are factors of $12$. The number $12$ is called the product.

Think of factors as the numbers you can split a group into with equal shares and no leftovers.

Example: Factors of 12

You can make $12$ in several ways:

So far, $1, 2, 3, 4, 6,$ and $12$ are all factors of $12$.

💡 Every number has at least two factors: $1$ and itself. For example, $1 \times 7 = 7$, so $1$ and $7$ are both factors of $7$.

Concept Check 🎯

The Division Test

How do you check whether a number is a factor? Use the division test:

🔑 Division Test: A number is a factor of a bigger number if it divides in evenly — the remainder is $0$.

Example: Is $4$ a factor of $20$?

$20 \div 4 = 5 \text{ with remainder } 0$

The remainder is $0$, so yes, $4$ is a factor of $20$. ✓

Example: Is $6$ a factor of $20$?

$20 \div 6 = 3 \text{ with remainder } 2$

The remainder is not $0$, so no, $6$ is not a factor of $20$. ✗

⚠️ Watch out: "Divides evenly" does not mean the answer is an even number. It means there is no remainder. For example, $20 \div 5 = 4$ exactly, so $5$ is a factor of $20$ even though $5$ is odd.

Division Test Practice 🧮

Divide and enter the remainder for each. (If it divides evenly, the remainder is $0$.)

1) $24 \div 3 = ?$ remainder 2) $24 \div 5 = ?$ remainder 3) $30 \div 6 = ?$ remainder

Concept Check 🎯

You've Got the Foundation

You now know what a factor is and how to test for one:

• A factor divides a number evenly (remainder $0$).
• Every number has $1$ and itself as factors.
• "Evenly" means no remainder — not "an even number."

In Part 2, we'll organize factors into factor pairs so we can find them quickly and never miss one.

🧩 Factors and Factor Pairs

Part 2 of 5 — Factor Pairs

🔑 The Idea: A factor pair is two numbers that multiply together to make a target number. Listing factor pairs is the neatest way to find all the factors of a number.

What Is a Factor Pair?

A factor pair of a number is two whole numbers whose product is that number.

For $12$, the factor pairs are:

$1 \times 12 \qquad 2 \times 6 \qquad 3 \times 4$

Each pair multiplies to $12$, so the factor pairs of $12$ are:

💡 Picture it: A factor pair tells you how to arrange objects into equal rows. $12$ stickers can go in $3$ rows of $4$, or $2$ rows of $6$, or $1$ row of $12$. Each arrangement is one factor pair.

A System for Finding Factor Pairs

To find all the factor pairs without missing any, count up from $1$ and test each number:

🔑 The Climbing Method: Start at $1$ and check $1, 2, 3, 4, \dots$ in order. Each time a number divides evenly, write down the pair. Stop when the two numbers in your pair meet or cross.

Example: Factor pairs of $18$

The factor pairs of $18$ are $1 \times 18$, $2 \times 9$, and $3 \times 6$.

Concept Check 🎯

Find the Missing Partner 🧮

Each line shows one number from a factor pair. Enter the partner that completes the pair.

1) $2 \times ? = 16$ 2) $5 \times ? = 35$ 3) $6 \times ? = 42$

Build the Factor Pairs of 30 🔽

Use the climbing method. Pick the correct partner for each factor of $30$.

🧩 Factors and Factor Pairs

Part 3 of 5 — Listing ALL the Factors

🔑 Goal: Use factor pairs to write the complete list of every factor of a number, in order, without missing any or repeating any.

From Factor Pairs to a Full Factor List

Once you have the factor pairs, just read off both numbers in each pair to get the full list of factors.

Example: All factors of $24$

Now list them smallest to largest:

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

So $24$ has $8$ factors.

💡 Tidy trick: Write the small partner of each pair on the left moving right, and the big partner on the right moving left. The two lists meet in the middle and you have every factor in order: $\underbrace{1,\ 2,\ 3,\ 4}_{\text{small ends}} \quad \underbrace{6,\ 8,\ 12,\ 24}_{\text{big ends}}$

Concept Check 🎯

Square Numbers: When a Factor Pairs With Itself

Some numbers have a factor pair where both numbers are the same.

Example: Factors of $36$

Because $6 \times 6 = 36$, the number $6$ appears only once in the list:

$1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\ 36$

🔑 Square numbers like $4, 9, 16, 25,$ and $36$ have a factor that multiplies by itself. Write that "middle" factor only once.

Count the Factors 🧮

Find every factor, then enter how many factors each number has.

1) Factors of $10$ → how many? (list: $1, 2, 5, 10$) 2) Factors of $9$ → how many? (remember $3 \times 3$) 3) Factors of $15$ → how many?

Put the Factors of 40 in Order 🔽

The factors of $40$ are $1, 2, 4, 5, 8, 10, 20, 40$. Fill in the missing factors in order from smallest to largest.

🧩 Factors and Factor Pairs

Part 4 of 5 — Prime, Composite & Real-World Factors

🔑 Big Idea: The number of factors a number has sorts it into two important groups — prime and composite — and factor pairs help us solve real grouping problems.

Prime and Composite Numbers

Counting the factors of a number tells you which kind it is:

Examples

• $7$ is prime: its only factors are $1$ and $7$.
• $12$ is composite: it has $1, 2, 3, 4, 6, 12$ — six factors.

⚠️ Special case: The number $1$ is neither prime nor composite. It has only one factor (itself), not two.

💡 Smallest prime fact: $2$ is the only even prime number. Every other even number has $2$ as a factor plus $1$ and itself, so it's composite.

Concept Check 🎯

Prime or Composite? 🔽

Decide whether each number is prime or composite.

Factors in the Real World: Arrays & Equal Groups

Factor pairs answer everyday questions about arranging things in equal rows.

Example: Arranging chairs

You have $20$ chairs and want equal rows. Every factor pair of $20$ gives a way to do it:

Because $3$ is not a factor of $20$, you cannot make exactly $3$ equal rows — someone's row would come up short!

💡 Word-problem clue: Whenever a problem says "equal groups," "equal rows," or "share evenly with none left over," it's really asking you to find factors.

Equal Groups Problems 🧮

1) A baker puts $18$ muffins into rows with $6$ muffins in each row. How many rows? $18 \div 6 = ?$ 2) A teacher splits $28$ students into $4$ equal teams. How many students per team? $28 \div 4 = ?$ 3) Can $24$ apples be shared evenly among $5$ baskets? Enter the remainder of $24 \div 5$. (If it's not $0$, they cannot share evenly.)

🧩 Factors and Factor Pairs

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) test whether a number is a factor, (2) find factor pairs, (3) list all factors in order, and (4) tell prime from composite. Let's put it all together.

Quick Reference

⚠️ Don't forget: $1$ and the number itself are always factors. And $1$ is neither prime nor composite.

Mixed Practice 🎯

Mixed Practice 🧮

1) Enter the missing partner: $7 \times ? = 56$ 2) How many factors does $14$ have? (list them in your head: $1, 2, 7, 14$) 3) Is $3$ a factor of $27$? Enter the remainder of $27 \div 3$. (remainder $0$ means yes)

Exit Quiz ✅

Answer all three to finish the lesson.