# Factors and Multiples: Two Sides of Multiplication

Remember from Lesson 1: factor × factor = product. Today we go deeper into the world of factors, and we meet their cousins, multiples.

## Factors: Dividing IN

The factors of a number are all the whole numbers that divide into it evenly, with no remainder. Another way to say it: factors are the whole numbers you can multiply together to get that number.

The best way to find all the factors is to look for factor pairs, two numbers that multiply together to give you the original number.

Example: Find all factor pairs of 24.

Start with 1 and work your way up:

| Factor Pair | Check | |---|---| | 1 × 24 | 1 × 24 = 24 ✓ | | 2 × 12 | 2 × 12 = 24 ✓ | | 3 × 8 | 3 × 8 = 24 ✓ | | 4 × 6 | 4 × 6 = 24 ✓ | | 5 × ? | 24 ÷ 5 = 4.8, not a whole number, so 5 is NOT a factor | | 6 × 4 | We already have this pair, so stop here |

So the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

## Multiples: Multiplying OUT

The multiples of a number are what you get when you multiply it by 1, 2, 3, 4, and so on. Multiples keep going forever. There is no last one.

Example: Multiples of 6: 6 × 1 = 6 → 6 × 2 = 12 → 6 × 3 = 18 → 6 × 4 = 24 → 6 × 5 = 30 → 6 × 6 = 36...

So the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42... (and they never stop)

> Key insight: Factors divide INTO a number. Multiples are what you get when you multiply OUT from a number.

## Common Factors and the GCF

When two numbers share the same factor, we call it a common factor. Look at the factors of 12 and 18 side by side:

` Factors of 12: 1 2 3 4 6 12 | | | | 1 2 3 6 ← COMMON FACTORS | | | | Factors of 18: 1 2 3 6 9 18 `

The common factors of 12 and 18 are: 1, 2, 3, and 6

The Greatest Common Factor (GCF) is the largest of those shared factors. The GCF of 12 and 18 is 6.

# Prime and Composite: The Building Blocks

Now that you can find factors, you are ready to sort numbers into two important groups: prime and composite.

## Prime Numbers

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Prime numbers are the atoms of mathematics. They cannot be broken down into smaller factor pairs.

Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23...

Look at why 7 is prime: - Can 7 be divided by 2? 7 ÷ 2 = 3.5, no. - Can 7 be divided by 3? 7 ÷ 3 = 2.33..., no. - Can 7 be divided by 4, 5, or 6? No. - The only factor pair is 1 × 7. Only two factors, so it is prime.

> Note: 2 is the only even prime number. Every other even number can be divided by 2, which means it has more than two factors.

## Composite Numbers

A composite number is a whole number greater than 1 that has more than two factors. Composite numbers CAN be broken down into smaller factor pairs.

Examples: - 4 → factors: 1, 2, 4 (three factors, composite) - 6 → factors: 1, 2, 3, 6 (four factors, composite) - 8 → factors: 1, 2, 4, 8 (four factors, composite) - 9 → factors: 1, 3, 9 (three factors, composite)

## The Special Case: 1

The number 1 is neither prime nor composite. It has only one factor (itself), so it does not fit either definition. This is a common trick question, so remember it!

## Factor Trees

A factor tree lets you break a composite number all the way down to its prime building blocks. This is called prime factorization.

Example: Factor tree for 36

` 36 / \ 6 6 / \ / \ 2 3 2 3 `

Prime factorization of 36: 2 × 2 × 3 × 3, which we write as 2² × 3²

Every path down the tree ends at a prime number. Those primes are the "atoms" that make up 36.

> Connection: When you simplify fractions in Lesson 3, you will be finding common factors. Factor trees help you find ALL the factors, which makes simplifying much easier.

# LCM: Where Multiples Meet

The Least Common Multiple (LCM) is the smallest multiple that two numbers share, the first place their multiples "meet."

## How to Find the LCM

The most reliable method: list the multiples of each number until you find the first one they have in common.

Example: Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20, 24... Multiples of 6: 6, 12, 18, 24, 30...

The first multiple both lists share is 12. So the LCM of 4 and 6 is 12.

## A Real-World LCM Problem

Imagine this: you wash your gym clothes every 3 days. Your friend washes theirs every 4 days. If you both wash today, in how many days will you both wash on the same day again?

- Your wash days: 3, 6, 9, 12, 15... - Your friend's wash days: 4, 8, 12, 16...

You both land on day 12. The answer is the LCM of 3 and 4, which is 12 days.

You already think about least common multiples in real life. You just did not have the math name for it yet.

# Place Value: The Position of Power

In our number system, the value of a digit depends on its position. The digit 4 means something very different in 400, in 40, in 4, and in 0.04. This is what makes our number system so powerful. Ten symbols (0-9) can represent any number ever conceived, just by changing position.

## The Place Value Chart

Every position in a number has a name and a value:

| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | |-----------|----------|------|------|---|--------|------------|-------------| | 1,000 | 100 | 10 | 1 | . | 0.1 | 0.01 | 0.001 |

Key pattern: Each place is 10 times the value of the place to its right, and 1/10 of the value of the place to its left.

- 1,000 is 10 times 100 - 100 is 10 times 10 - 10 is 10 times 1 - 1 is 10 times 0.1 - 0.1 is 10 times 0.01 - ...and so on

## Decimals: Extending to the Right

The decimal point is the anchor. Everything to the left is a whole number place. Everything to the right is a fraction of 1:

- Tenths (0.1): One-tenth of a whole. Ten of them make 1. - Hundredths (0.01): One-hundredth of a whole. One hundred of them make 1. - Thousandths (0.001): One-thousandth of a whole. One thousand of them make 1.

## Real-World Decimals: Money

You already use decimal place value every time you work with money. Money is a decimal system:

**$4.75** = 4 ones + 7 tenths + 5 hundredths = 4 dollars and 75 cents A penny is one-hundredth of a dollar, which is why it is written as $0.01 because the 1 is in the hundredths place.

## Reading Decimals Correctly

To read a decimal number correctly, the last digit's place value gives you the name of the whole decimal:

- 0.3 → "three tenths" (last digit is in tenths) - 0.47 → "forty-seven hundredths" (last digit is in hundredths) - 0.375 → "three hundred seventy-five thousandths" (last digit is in thousandths) - 3,572.49 → "three thousand five hundred seventy-two and forty-nine hundredths"

> Bridge to Lesson 3: Now that you understand place value, the next lesson connects decimals to fractions, because they are two different ways of writing the same numbers. 0.375 and 375/1000 are identical in value.