Factors and Multiples

Factors and multiples are two sides of the same coin. If you can multiply $4 \times 6 = 24$, then $4$ and $6$ are factors of $24$, and $24$ is a multiple of both $4$ and $6$. Understanding this relationship — along with quick divisibility tests — is the foundation for simplifying fractions, finding common denominators, and solving real-world problems that involve splitting things into equal groups or finding shared schedules.

What Are Factors?

A factor of a number divides into it evenly — with no remainder.

$\text{If } a \div b = \text{whole number (no remainder), then } b \text{ is a factor of } a.$

Every whole number greater than $1$ has at least two factors: $1$ and itself.

Listing All Factors

To find all factors of a number, check each whole number starting from $1$ to see if it divides evenly. You can stop checking when you reach a number whose square exceeds the original — beyond that point, any factor you find would have a smaller partner you already discovered.

Example 1: Find All Factors of 36

Start with $1$ and work up:

• $36 \div 1 = 36$ — both $1$ and $36$ are factors
• $36 \div 2 = 18$ — both $2$ and $18$ are factors
• $36 \div 3 = 12$ — both $3$ and $12$ are factors
• $36 \div 4 = 9$ — both $4$ and $9$ are factors
• $36 \div 5 = 7.2$ — not a whole number, so $5$ is not a factor
• $36 \div 6 = 6$ — $6$ is a factor (paired with itself)

Stop here because $7^2 = 49 > 36$.

Factors of 36: $\{1, 2, 3, 4, 6, 9, 12, 18, 36\}$

Factor Pairs

Factors come in pairs that multiply to give the original number:

$36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6$

When a number is a perfect square (like $36$), one factor pair consists of the same number repeated ($6 \times 6$).

Example 2: Find All Factors of 28

• $28 \div 1 = 28$ — factors: $1, 28$
• $28 \div 2 = 14$ — factors: $2, 14$
• $28 \div 3 = 9.33\ldots$ — not a factor
• $28 \div 4 = 7$ — factors: $4, 7$
• $28 \div 5 = 5.6$ — not a factor

Stop at $6$ because $6^2 = 36 > 28$.

Factors of 28: $\{1, 2, 4, 7, 14, 28\}$

Divisibility Rules

Divisibility rules let you quickly check whether a number is a factor without doing long division. These shortcuts save time and are especially useful with large numbers.

Divisible by 2

A number is divisible by $2$ if its last digit is even ($0, 2, 4, 6, 8$).

• $574$ — last digit is $4$ (even), so $574 \div 2 = 287$. Divisible.
• $831$ — last digit is $1$ (odd). Not divisible by $2$.

Divisible by 3

A number is divisible by $3$ if the sum of its digits is divisible by $3$.

• $123$: digit sum $= 1 + 2 + 3 = 6$. Since $6 \div 3 = 2$, yes, $123$ is divisible by $3$.
• $274$: digit sum $= 2 + 7 + 4 = 13$. Since $13 \div 3 = 4$ remainder $1$, no.

Divisible by 5

A number is divisible by $5$ if its last digit is $0$ or $5$.

• $3{,}405$ — last digit is $5$. Divisible.
• $782$ — last digit is $2$. Not divisible.

Divisible by 9

A number is divisible by $9$ if the sum of its digits is divisible by $9$. (Same test as the rule for $3$, but with $9$.)

• $738$: digit sum $= 7 + 3 + 8 = 18$. Since $18 \div 9 = 2$, yes.
• $514$: digit sum $= 5 + 1 + 4 = 10$. Since $10 \div 9 = 1$ remainder $1$, no.

Divisible by 10

A number is divisible by $10$ if its last digit is $0$.

• $4{,}560$ — last digit is $0$. Divisible.
• $4{,}565$ — last digit is $5$. Not divisible by $10$ (but divisible by $5$).

Example 3: Applying Divisibility Rules

Test whether $2{,}340$ is divisible by $2$, $3$, $5$, $9$, and $10$.

• By 2: Last digit is $0$ (even). Yes.
• By 3: Digit sum $= 2 + 3 + 4 + 0 = 9$. $9 \div 3 = 3$. Yes.
• By 5: Last digit is $0$. Yes.
• By 9: Digit sum $= 9$. $9 \div 9 = 1$. Yes.
• By 10: Last digit is $0$. Yes.

$2{,}340$ is divisible by all five.

What Are Multiples?

A multiple of a number is the product of that number and any positive whole number. Equivalently, multiples are what you get when you “count by” a number.

$\text{Multiples of } 7\!: \quad 7, 14, 21, 28, 35, 42, \ldots$

Every number has infinitely many multiples (you can always multiply by the next whole number).

Listing Multiples

To list the first several multiples of $n$, multiply $n$ by $1, 2, 3, 4, 5, \ldots$

Example 4: First Eight Multiples of 6

$6 \times 1 = 6$ $6 \times 2 = 12$ $6 \times 3 = 18$ $6 \times 4 = 24$ $6 \times 5 = 30$ $6 \times 6 = 36$ $6 \times 7 = 42$ $6 \times 8 = 48$

First eight multiples of 6: $6, 12, 18, 24, 30, 36, 42, 48$

The Relationship Between Factors and Multiples

Factors and multiples are inverse ideas:

• If $4$ is a factor of $20$, then $20$ is a multiple of $4$
• If $20$ is a multiple of $4$, then $4$ is a factor of $20$

They are two ways of describing the same division relationship: $20 \div 4 = 5$ with no remainder.

Quick Comparison

Real-World Application: Carpentry — Cutting Equal Lengths

A carpenter has a board that is $72$ inches long and needs to cut it into equal pieces with no waste. What lengths are possible?

The possible piece lengths are the factors of 72:

$\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}$

If the project calls for pieces between $8$ and $12$ inches, the carpenter can cut $8$-inch pieces ($72 \div 8 = 9$ pieces), $9$-inch pieces ($72 \div 9 = 8$ pieces), or $12$-inch pieces ($72 \div 12 = 6$ pieces). Knowing factors lets you plan cuts without trial and error.

Real-World Application: Retail — Packaging Products

A warehouse needs to pack $180$ items into boxes of equal size. The manager wants each box to hold between $10$ and $20$ items. Which box sizes work?

Find factors of $180$ between $10$ and $20$:

• $180 \div 10 = 18$ — $10$ is a factor, giving $18$ boxes
• $180 \div 12 = 15$ — $12$ is a factor, giving $15$ boxes
• $180 \div 15 = 12$ — $15$ is a factor, giving $12$ boxes
• $180 \div 18 = 10$ — $18$ is a factor, giving $10$ boxes
• $180 \div 20 = 9$ — $20$ is a factor, giving $9$ boxes

Possible box sizes: $10$, $12$, $15$, $18$, or $20$ items per box. The manager picks the size that best matches the shipping cartons available.

Common Mistakes to Avoid

1. Confusing factors with multiples. Factors are smaller than or equal to the number. Multiples are larger than or equal to the number. $3$ is a factor of $12$, but $12$ is a multiple of $3$ — not the other way around.
2. Forgetting $1$ and the number itself. Every number greater than $1$ has at least two factors: $1$ and the number itself. Do not skip them when listing.
3. Stopping the factor search too early. Check up to the square root of the number. For $100$, check through $10$ (since $10 \times 10 = 100$).
4. Misapplying the digit-sum rule. The digit-sum test works for divisibility by $3$ and $9$, but not for $2$, $5$, or $10$ (those use the last digit).
5. Thinking a number has finitely many multiples. Multiples go on forever — there is no “largest multiple” of any number.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A factor of a number divides into it with no remainder — list them by testing systematically up to the square root
• Factor pairs multiply to give the original number — they come in pairs (e.g., $3 \times 12 = 36$)
• A multiple of a number is the result of multiplying it by a positive whole number — multiples are infinite
• Divisibility rules provide quick checks: last digit for $2$, $5$, $10$; digit sum for $3$ and $9$
• Factors and multiples are inverse concepts: if $a$ is a factor of $b$, then $b$ is a multiple of $a$
• Real-world uses include cutting materials into equal pieces, packaging items evenly, and scheduling repeated events

Return to Pre-Algebra for more topics in this section.