Factors, Multiples, and Divisibility

Have you ever split a pizza evenly among friends, or wondered why some numbers just feel “cleaner” to work with than others? Maybe you’ve noticed that dividing by 10 is easy, but dividing by 7 feels awkward. There’s a reason for that, and understanding it will unlock a whole new way of seeing numbers.

If terms like “prime factorization” or “least common multiple” sound intimidating, take a breath. You’ve actually been working with these concepts your whole life - every time you’ve made change, shared snacks equally, or figured out when two schedules line up. We’re just going to give you the vocabulary and tools to do it more deliberately.

Core Concepts

What Are Factors?

A factor of a number is any whole number that divides into it evenly - meaning there’s no remainder. Think of factors as the building blocks that multiply together to make a number.

For example, the factors of 12 are: 1, 2, 3, 4, 6, and 12. Each of these divides into 12 with nothing left over.

You already use factors without realizing it. When you think “I can split this $12 evenly among 3 people, giving each person $4,” you’ve just identified that 3 and 4 are factors of 12.

What Are Multiples?

Multiples are the opposite of factors. A multiple of a number is what you get when you multiply that number by any whole number.

The multiples of 5 are: 5, 10, 15, 20, 25, 30, and so on forever.

You encounter multiples constantly: nickels (multiples of 5 cents), dozens (multiples of 12), hours on a clock (multiples of the hour hand position).

What Is Divisibility?

Divisibility just means “can be divided evenly.” When we say “36 is divisible by 9,” we mean 36 divided by 9 equals a whole number (4) with no remainder.

Divisibility is the bridge between factors and multiples:

• If $a$ is divisible by $b$, then $b$ is a factor of $a$
• If $a$ is divisible by $b$, then $a$ is a multiple of $b$

Notation and Terminology

The vertical bar notation $a \mid b$ means “$a$ divides $b$” or “$b$ is divisible by $a$.” So $3 \mid 12$ is a true statement.

Key Properties and Rules

Divisibility Rules

These rules let you quickly check if a number is divisible by common values - no calculator required. They’re like shortcuts your brain can use.

Pro tip: The rule for 3 and 9 works because of how our number system is built. Each place value (10, 100, 1000…) is always one more than a multiple of 9. This means when you add up the digits, you’re essentially finding the remainder when dividing by 9 - clever, right?

Finding All Factors of a Number

Here’s a systematic approach that ensures you never miss a factor:

1. Start with 1 and the number itself (they’re always factors)
2. Check 2, 3, 4, and so on
3. For each factor you find, there’s a “partner” factor (the result of the division)
4. Stop when your factors start repeating (when you reach the square root)

Prime vs. Composite Numbers

• Prime numbers can only be divided evenly by 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23…
• Composite numbers have additional factors: 4, 6, 8, 9, 10, 12, 14, 15, 16…
• Special cases: 1 is neither prime nor composite. 2 is the only even prime number.

Prime numbers are the “atoms” of arithmetic - every whole number greater than 1 is either prime or can be built by multiplying primes together.

Prime Factorization with Factor Trees

A factor tree breaks a number down into its prime building blocks. Here’s how to build one:

1. Write your number at the top
2. Find any two factors (not 1 and the number itself)
3. Branch down to those two factors
4. If a factor is prime, circle it - you’re done with that branch
5. If a factor is composite, keep breaking it down
6. When all branches end in primes, collect them

        72
       /  \
      8    9
     /\    /\
    2  4  3  3
      /\
     2  2

72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

The amazing thing: no matter how you build your factor tree, you’ll always get the same prime factors. This is called the Fundamental Theorem of Arithmetic.

Greatest Common Factor (GCF)

The GCF is the largest number that divides evenly into two or more numbers. It’s incredibly useful for simplifying fractions.

Method 1: List all factors

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12
• Greatest common factor: 12

Method 2: Use prime factorization

• $24 = 2^3 \times 3$
• $36 = 2^2 \times 3^2$
• GCF = Take the lowest power of each common prime: $2^2 \times 3^1 = 4 \times 3 = 12$

Least Common Multiple (LCM)

The LCM is the smallest positive number that is a multiple of two or more numbers. It’s essential for adding fractions with different denominators.

Method 1: List multiples

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28…
• Multiples of 6: 6, 12, 18, 24, 30…
• First common multiple: 12

Method 2: Use prime factorization

• $4 = 2^2$
• $6 = 2 \times 3$
• LCM = Take the highest power of each prime that appears: $2^2 \times 3 = 4 \times 3 = 12$

Handy relationship: For any two numbers $a$ and $b$: $$\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$$

Examples

       180
      /   \
     10    18
    / \    / \
   2   5  2   9
             / \
            3   3

Real-World Applications

Simplifying Fractions

When you reduce $\frac{48}{60}$ to lowest terms, you’re dividing both numbers by their GCF:

$$\text{GCF}(48, 60) = 12$$ $$\frac{48}{60} = \frac{48 \div 12}{60 \div 12} = \frac{4}{5}$$

Adding Fractions with Different Denominators

To add $\frac{5}{6} + \frac{3}{8}$, you need a common denominator - ideally the LCM:

$$\text{LCM}(6, 8) = 24$$ $$\frac{5}{6} = \frac{20}{24} \quad \text{and} \quad \frac{3}{8} = \frac{9}{24}$$ $$\frac{5}{6} + \frac{3}{8} = \frac{20}{24} + \frac{9}{24} = \frac{29}{24}$$

Splitting Things Equally

You have 36 cupcakes and 48 cookies to divide among party bags so each bag gets the same combination, with nothing left over. What’s the maximum number of bags you can make?

The GCF of 36 and 48 is 12. You can make 12 bags, each with 3 cupcakes and 4 cookies.

When Events Align

Two traffic lights both turn green together. One has a 40-second cycle, the other has a 60-second cycle. When will they both turn green at the same time again?

Find the LCM: $\text{LCM}(40, 60) = 120$ seconds = 2 minutes

Password/Combination Locks

Many number puzzles and codes rely on prime factorization. Understanding factors helps you see patterns in lock combinations, clock arithmetic, and cyclic systems.

Self-Test Problems

Summary

• Factors divide evenly into a number; multiples are what you get when you multiply a number
• Divisibility rules are mental shortcuts: check if a number ends in 0/5 (divisible by 5), add digits for 3 and 9, check last two digits for 4
• Prime numbers have exactly two factors (1 and themselves); composite numbers have more than two
• Prime factorization breaks any number into its prime building blocks - like finding the “atoms” of a number
• GCF (Greatest Common Factor): Use lowest powers of common primes. Helpful for simplifying fractions and dividing things equally
• LCM (Least Common Multiple): Use highest powers of all primes. Essential for finding common denominators and scheduling problems
• Remember: $\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$ - a great way to check your work

These concepts are the foundation for working with fractions, ratios, and algebraic expressions. Once you’re comfortable finding factors and multiples, you’ll find that many math problems become much more manageable.