Number Theory and Divisibility

Lesson: Factors and Multiples

Numbers can be connected in different ways. Two important number ideas are factors and multiples. Learning these helps you understand division, multiplication, fractions, and many other math skills.

In this lesson, you will learn what factors are, what multiples are, how they are different, and how to find them.

What is a factor?

A factor of a number is a whole number that divides evenly into that number. If there is no remainder, then the number is a factor.

For example, 3 is a factor of 12 because $$12 \div 3 = 4$$ and there is no remainder.

You can also think about factors with multiplication. If two numbers multiply to make a product, then both numbers are factors of the product.

For example, since $$3 \times 4 = 12,$$ both 3 and 4 are factors of 12.

What is a multiple?

A multiple of a number is what you get when you multiply that number by a whole number.

For example, multiples of 5 are found by multiplying 5 by 1, 2, 3, 4, and so on:

• \(5 \times 1 = 5\)
• \(5 \times 2 = 10\)
• \(5 \times 3 = 15\)
• \(5 \times 4 = 20\)

So some multiples of 5 are 5, 10, 15, 20.

Factors and multiples are different

It is easy to mix them up, so let’s compare them.

• Factors are numbers you can divide by evenly.
• Multiples are numbers you get when you multiply.

For the number 12:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Some multiples of 12: 12, 24, 36, 48, 60

Notice something important:

• A number has only a limited number of factors.
• A number has infinitely many multiples.

How to find the factors of a number

To find factors, ask: What multiplication facts make this number?

Let’s find the factors of 18. We look for pairs of numbers that multiply to 18.

• \(1 \times 18 = 18\)
• \(2 \times 9 = 18\)
• \(3 \times 6 = 18\)

So the factors of 18 are 1, 2, 3, 6, 9, 18.

A helpful idea is to find factor pairs. A factor pair is two numbers that multiply to make the number.

How to find multiples of a number

To find multiples, multiply the number by 1, 2, 3, 4, and so on.

For 7:

• \(7 \times 1 = 7\)
• \(7 \times 2 = 14\)
• \(7 \times 3 = 21\)
• \(7 \times 4 = 28\)
• \(7 \times 5 = 35\)

So some multiples of 7 are 7, 14, 21, 28, 35.

Important facts to remember

• Every number has 1 as a factor.
• Every number is a factor of itself.
• Every number is a multiple of 1.
• The first multiple of a number is the number itself.

For example, for 9:

• Factors of 9: 1, 3, 9
• Some multiples of 9: 9, 18, 27, 36

Worked Example 1: Find the factors of 16

Step 1: Look for multiplication facts that make 16.

• \(1 \times 16 = 16\)
• \(2 \times 8 = 16\)
• \(4 \times 4 = 16\)

Step 2: List all the factors.

The factors of 16 are 1, 2, 4, 8, 16.

Worked Example 2: Find the first 5 multiples of 6

Multiply 6 by 1 through 5:

• \(6 \times 1 = 6\)
• \(6 \times 2 = 12\)
• \(6 \times 3 = 18\)
• \(6 \times 4 = 24\)
• \(6 \times 5 = 30\)

So the first 5 multiples of 6 are 6, 12, 18, 24, 30.

Worked Example 3: Is 5 a factor of 35? Is 35 a multiple of 5?

Check by dividing:

$$35 \div 5 = 7$$

There is no remainder, so 5 is a factor of 35.

Now think about multiplication:

$$5 \times 7 = 35$$

So 35 is a multiple of 5.

This shows that factors and multiples are connected.

Worked Example 4: List the factors of 24 and the first 4 multiples of 24

First, find factor pairs of 24:

• \(1 \times 24 = 24\)
• \(2 \times 12 = 24\)
• \(3 \times 8 = 24\)
• \(4 \times 6 = 24\)

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

Now find the first 4 multiples of 24:

• \(24 \times 1 = 24\)
• \(24 \times 2 = 48\)
• \(24 \times 3 = 72\)
• \(24 \times 4 = 96\)

So the first 4 multiples of 24 are 24, 48, 72, 96.

A quick way to tell the difference

• If you are asking, “What numbers go into this number evenly?” you are finding factors.
• If you are asking, “What numbers do I get by skip-counting or multiplying?” you are finding multiples.

For example:

• Factors of 10: 1, 2, 5, 10
• Some multiples of 10: 10, 20, 30, 40, 50

Common mistakes to avoid

• Do not list numbers that do not divide evenly as factors.
• Do not confuse factor pairs with multiples.
• Remember that factors are usually smaller than or equal to the number, but multiples can keep getting bigger.

For example, 5 is not a factor of 12 because $$12 \div 5$$ does not make a whole number.

Let’s review

A factor divides a number evenly. A multiple is found by multiplying a number by whole numbers.

If you know your multiplication facts well, you can find factors and multiples more easily. Factors help you break numbers apart, and multiples help you build numbers up.

Summary

• Factors are numbers that divide evenly into another number.
• Multiples are numbers made by multiplying a number by whole numbers.
• Factors can be found using multiplication pairs.
• Multiples can be found by skip-counting or multiplying.
• Factors and multiples are connected, but they are not the same.

Keep practicing by asking yourself: Am I dividing, or am I multiplying? That will help you decide whether to look for factors or multiples.

Prime and Composite Numbers

Numbers can be sorted into groups in different ways. One important way is by looking at their factors.

A factor of a number is a whole number that divides the number evenly, with no remainder. For example, the factors of \(12\) are \(1, 2, 3, 4, 6, 12\).

When we know a number’s factors, we can decide whether it is prime, composite, or a special number like \(1\).

What is a prime number?

A prime number is a whole number greater than \(1\) that has exactly two distinct factors: \(1\) and itself.

Examples of prime numbers are \(2, 3, 5, 7, 11\), and \(13\).

Let’s look at \(7\):

• \(1 \times 7 = 7\)
• No other whole numbers divide \(7\) evenly.

So \(7\) has exactly two distinct factors: \(1\) and \(7\). That means \(7\) is prime.

What is a composite number?

A composite number is a whole number greater than \(1\) that has more than two distinct factors.

Examples of composite numbers are \(4, 6, 8, 9, 10\), and \(12\).

Let’s look at \(8\):

• \(1 \times 8 = 8\)
• \(2 \times 4 = 8\)

The factors of \(8\) are \(1, 2, 4, 8\). Since \(8\) has more than two distinct factors, it is composite.

The special case: number \(1\)

The number \(1\) is not prime and not composite.

Why? Because \(1\) has only one factor: itself.

To be prime, a number must have exactly two distinct factors. To be composite, a number must have more than two distinct factors. The number \(1\) does not fit either rule.

Important facts to remember

• Prime numbers are greater than \(1\).
• Prime numbers have exactly two distinct factors.
• Composite numbers are greater than \(1\).
• Composite numbers have more than two distinct factors.
• The number \(1\) is neither prime nor composite.
• The number \(2\) is the smallest prime number.

Why is \(2\) special?

The number \(2\) has exactly two distinct factors: \(1\) and \(2\). So it is prime.

It is also the only even prime number. Every other even number is divisible by \(2\), so it has at least three factors: \(1\), \(2\), and itself. That makes those even numbers composite.

How to tell if a number is prime or composite

1. Make sure the number is greater than \(1\).
2. Find all the whole-number factors.
3. Count the distinct factors.
4. If there are exactly two, the number is prime.
5. If there are more than two, the number is composite.

You can also think about factor pairs. A factor pair is two whole numbers multiplied together to make a product.

For example, the factor pairs of \(12\) are:

Since \(12\) has more than one factor pair, it has more than two factors, so it is composite.

If a number has only one factor pair, \(1 \times \text{that number}\), then it is prime.

Worked Example 1: Is \(5\) prime or composite?

Step 1: Find the factors of \(5\).

• \(1\) divides \(5\) evenly.
• \(5\) divides \(5\) evenly.

The factors are \(1\) and \(5\).

Step 2: Count the distinct factors.

There are exactly \(2\) distinct factors.

Answer: \(5\) is prime.

Worked Example 2: Is \(9\) prime or composite?

Step 1: Find the factors of \(9\).

• \(1 \times 9 = 9\)
• \(3 \times 3 = 9\)

The distinct factors are \(1, 3, 9\).

Step 2: Count the distinct factors.

There are \(3\) distinct factors.

Answer: \(9\) is composite.

Worked Example 3: Is \(1\) prime or composite?

Step 1: Find the factors of \(1\).

• The only factor is \(1\).

Step 2: Count the distinct factors.

There is only \(1\) factor.

Answer: \(1\) is neither prime nor composite.

Worked Example 4: Is \(21\) prime or composite?

Step 1: Look for factor pairs.

• \(1 \times 21 = 21\)
• \(3 \times 7 = 21\)

So the factors of \(21\) are \(1, 3, 7, 21\).

Step 2: Count the distinct factors.

There are \(4\) distinct factors.

Answer: \(21\) is composite.

Prime and composite numbers from \(1\) to \(20\)

• Prime: \(2, 3, 5, 7, 11, 13, 17, 19\)
• Composite: \(4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20\)
• Neither: \(1\)

A helpful tip

If a number is even and greater than \(2\), it is always composite. That is because it can be divided evenly by \(2\).

For example:

So \(14\) has factors \(1, 2, 7, 14\), which means it is composite.

Common mistakes to avoid

• Do not say \(1\) is prime. It has only one factor.
• Do not forget to count distinct factors.
• Do not assume all odd numbers are prime. For example, \(9\) and \(15\) are odd, but they are composite.

Summary

Prime and composite numbers are classified by their factors.

• A prime number has exactly two distinct factors: \(1\) and itself.
• A composite number has more than two distinct factors.
• The number \(1\) is neither prime nor composite.

When you are unsure, list the factors or factor pairs. Then count how many distinct factors the number has.

Divisibility Rules help us tell whether one number can be divided by another number evenly, without doing the full division.

If a number divides evenly, it means there is no remainder.

For example, \(24 \div 6 = 4\), so 24 is divisible by 6. But \(25 \div 6\) does not divide evenly, so 25 is not divisible by 6.

Learning divisibility rules is useful because it helps you work faster with multiplication, division, fractions, and patterns in numbers.

Here are the divisibility rules you need to know:

• Divisible by 2: The number ends in 0, 2, 4, 6, or 8.
• Divisible by 3: Add the digits. If the sum is divisible by 3, then the number is divisible by 3.
• Divisible by 4: Look at the last 2 digits. If that 2-digit number is divisible by 4, then the whole number is divisible by 4.
• Divisible by 5: The number ends in 0 or 5.
• Divisible by 6: The number must be divisible by both 2 and 3.
• Divisible by 9: Add the digits. If the sum is divisible by 9, then the number is divisible by 9.
• Divisible by 10: The number ends in 0.

Let’s look at each rule more closely.

Rule for 2

Numbers divisible by 2 are called even numbers. If the ones digit is 0, 2, 4, 6, or 8, the number can be divided by 2 evenly.

Examples: 18, 42, 90, and 136 are all divisible by 2.

Rule for 3

Add all the digits in the number. If that total can be divided by 3 evenly, then the original number is divisible by 3.

Example: For 123, add \(1 + 2 + 3 = 6\). Since 6 is divisible by 3, 123 is divisible by 3.

Rule for 4

Look only at the last two digits. If those two digits make a number divisible by 4, then the whole number is divisible by 4.

Example: In 316, the last two digits are 16. Since \(16 \div 4 = 4\), 316 is divisible by 4.

Rule for 5

If the number ends in 0 or 5, it is divisible by 5.

Examples: 35, 70, and 145 are divisible by 5.

Rule for 6

A number is divisible by 6 if it passes both the rule for 2 and the rule for 3.

That means the number must be even, and the sum of its digits must be divisible by 3.

Example: For 42, it ends in 2, so it is divisible by 2. Also, \(4 + 2 = 6\), and 6 is divisible by 3. So 42 is divisible by 6.

Rule for 9

Add the digits. If the sum is divisible by 9, then the number is divisible by 9.

Example: For 261, add \(2 + 6 + 1 = 9\). Since 9 is divisible by 9, 261 is divisible by 9.

Rule for 10

If the number ends in 0, it is divisible by 10.

Examples: 40, 120, and 3,560 are divisible by 10.

Worked Example 1

Is 58 divisible by 2, 5, or 10?

1. For 2: 58 ends in 8, so yes, it is divisible by 2.
2. For 5: 58 does not end in 0 or 5, so no, it is not divisible by 5.
3. For 10: 58 does not end in 0, so no, it is not divisible by 10.

So, 58 is divisible by 2 only.

Worked Example 2

Is 234 divisible by 3, 6, and 9?

1. Add the digits: \(2 + 3 + 4 = 9\).
2. Since 9 is divisible by 3, 234 is divisible by 3.
3. Since 234 ends in 4, it is divisible by 2.
4. Because it is divisible by both 2 and 3, it is divisible by 6.
5. Since the digit sum is 9, it is also divisible by 9.

So, 234 is divisible by 3, 6, and 9.

Worked Example 3

Is 412 divisible by 4?

1. Look at the last two digits: 12.
2. Since \(12 \div 4 = 3\), 12 is divisible by 4.
3. So, 412 is divisible by 4.

Worked Example 4

Which of these numbers are divisible by 2, 3, 5, 6, 9, and 10: \(180\)?

1. For 2: 180 ends in 0, so yes.
2. For 3: \(1 + 8 + 0 = 9\), so yes.
3. For 5: it ends in 0, so yes.
4. For 6: it is divisible by both 2 and 3, so yes.
5. For 9: the digit sum is 9, so yes.
6. For 10: it ends in 0, so yes.

So, 180 is divisible by 2, 3, 5, 6, 9, and 10.

Helpful Tips

• For 3 and 9, always add the digits.
• For 4, only check the last two digits.
• For 6, do not guess. Check both 2 and 3.
• For 2, 5, and 10, the last digit tells you the answer.

Common Mistakes to Avoid

• Thinking that a number divisible by 2 is always divisible by 6. It must also be divisible by 3.
• Forgetting to add all the digits when checking 3 or 9.
• Checking the whole number for 4 instead of just the last two digits.
• Mixing up the rules for 5 and 10. Numbers divisible by 10 must end in 0, but numbers divisible by 5 can end in 0 or 5.

Quick Practice to Try

• Is 75 divisible by 3 and 5?
• Is 128 divisible by 4?
• Is 246 divisible by 6?
• Is 333 divisible by 9?

Summary

Divisibility rules are quick tests that help you know whether a number can be divided evenly. By checking the last digit, the last two digits, or the sum of the digits, you can test numbers for 2, 3, 4, 5, 6, 9, and 10 without doing full division.

The more you practice these rules, the faster and easier they will become.

Prime Factorization means breaking a number into a multiplication of prime numbers.

This is an important skill because it helps us understand how numbers are built. It also helps later with fractions, finding common factors, and other number problems.

Before learning prime factorization, we need to know two important ideas: prime numbers and composite numbers.

A prime number has exactly 2 factors: 1 and itself.

• Examples: 2, 3, 5, 7, 11

A composite number has more than 2 factors.

• Examples: 4, 6, 8, 9, 10, 12

The number 1 is not prime and not composite.

When we do prime factorization, we keep breaking a number apart until all the factors are prime numbers.

How to Find Prime Factorization

There are two common ways to do this:

1. Factor tree
2. Continuous division

Let’s learn both.

1. Factor Tree Method

A factor tree starts with the number you want to break apart. Then you split it into two factors. If a factor is composite, split it again. Stop when every branch ends in a prime number.

Important: It does not matter which pair of factors you choose first. If you keep going until all the numbers are prime, you will get the same prime factorization.

Worked Example 1: Prime Factorization of 12

Start with 12. One factor pair is 3 and 4.

Then break 4 into 2 and 2.

Now all the end numbers are prime:

• 3 is prime
• 2 is prime
• 2 is prime

So the prime factorization of 12 is:

$$12 = 2 \times 2 \times 3$$

We can also write this in order from least to greatest:

$$12 = 2 \times 2 \times 3$$

Worked Example 2: Prime Factorization of 18

Start with 18. One factor pair is 2 and 9.

Then break 9 into 3 and 3.

Now all factors are prime:

• 2 is prime
• 3 is prime
• 3 is prime

So the prime factorization of 18 is:

$$18 = 2 \times 3 \times 3$$

Notice that we do not stop at 9, because 9 is composite. We must keep going until every factor is prime.

2. Continuous Division Method

In this method, divide the number by prime numbers step by step.

It is often easiest to try the smallest prime numbers first: 2, then 3, then 5, then 7, and so on.

Keep dividing until the answer is 1.

Worked Example 3: Prime Factorization of 24

Let’s divide by prime numbers.

Since 24 is even, divide by 2:

$$24 \div 2 = 12$$

12 is even, so divide by 2 again:

$$12 \div 2 = 6$$

6 is even, so divide by 2 again:

$$6 \div 2 = 3$$

3 is prime, so divide by 3:

$$3 \div 3 = 1$$

The prime numbers we used are 2, 2, 2, and 3.

So the prime factorization of 24 is:

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

Worked Example 4: Prime Factorization of 45

45 is not even, so it cannot be divided by 2.

But 4 + 5 = 9, and 9 is divisible by 3, so 45 is divisible by 3.

Divide by 3:

$$45 \div 3 = 15$$

15 is also divisible by 3:

$$15 \div 3 = 5$$

5 is prime, so divide by 5:

$$5 \div 5 = 1$$

The prime numbers we used are 3, 3, and 5.

So the prime factorization of 45 is:

$$45 = 3 \times 3 \times 5$$

How to Check Your Answer

You can check a prime factorization in two ways:

1. Make sure every factor is prime.
2. Multiply the prime factors to see if you get the original number.

For example, for 18:

$$2 \times 3 \times 3 = 18$$

Since 2, 3, and 3 are all prime, the answer is correct.

Helpful Tips

• If a number is even, try dividing by 2 first.
• If a number ends in 0 or 5, try dividing by 5.
• If the digits add to a multiple of 3, try dividing by 3.
• Keep going until all factors are prime.
• Do not use 1 in a prime factorization.

Common Mistakes

• Stopping too soon: For example, writing \(12 = 3 \times 4\). This is not prime factorization because 4 is not prime.
• Using 1 as a prime factor: 1 is not prime.
• Missing a factor: Make sure the prime factors multiply back to the original number.

Why Different Factor Trees Still Work

You can start a factor tree in different ways.

For 12, you could do:

$$12 = 3 \times 4$$

or

$$12 = 2 \times 6$$

Then keep factoring:

$$4 = 2 \times 2$$

$$6 = 2 \times 3$$

Both ways give the same prime factors:

$$12 = 2 \times 2 \times 3$$

So even if your tree looks different, the final prime factorization will be the same.

Summary

Let’s review the big ideas.

• Prime factorization means writing a number as a product of prime numbers.
• A prime number has exactly 2 factors: 1 and itself.
• A composite number has more than 2 factors.
• You can find prime factorization using a factor tree or continuous division.
• Keep factoring until all factors are prime.
• Check your answer by multiplying the prime factors back together.

Examples:

• $$12 = 2 \times 2 \times 3$$
• $$18 = 2 \times 3 \times 3$$
• $$24 = 2 \times 2 \times 2 \times 3$$
• $$45 = 3 \times 3 \times 5$$

Prime factorization is like finding the building blocks of a number. Once you know those building blocks, you can understand numbers much better.

Greatest Common Factor (GCF) helps us find the biggest number that can divide two or more numbers evenly.

A factor of a number is a whole number that multiplies with another whole number to make that number.

For example, the factors of \(12\) are:

This is because:

• \(1 \times 12 = 12\)
• \(2 \times 6 = 12\)
• \(3 \times 4 = 12\)

When two numbers have some of the same factors, those are called common factors.

The greatest common factor is the largest factor they share.

Let’s learn how to find it step by step.

Why is GCF useful?

• It helps us simplify fractions.
• It helps us group objects into equal groups.
• It helps us understand how numbers are connected.

Method 1: List the factors

One easy way to find the GCF is to list all the factors of each number, then look for the biggest one they have in common.

Worked Example 1: Find the GCF of \(12\) and \(18\)

First, list the factors of each number.

Factors of \(12\):

Factors of \(18\):

Now find the common factors. These are the numbers in both lists:

The greatest one is \(6\).

So, the GCF of \(12\) and \(18\) is \(6\).

Worked Example 2: Find the GCF of \(24\) and \(36\)

List the factors of each number.

Factors of \(24\):

Factors of \(36\):

Common factors:

The greatest common factor is \(12\).

Method 2: Use factor pairs

You can also find factors by making factor pairs.

For example, factor pairs of \(20\) are:

• \(1 \times 20\)
• \(2 \times 10\)
• \(4 \times 5\)

So the factors of \(20\) are:

This method helps you make sure you do not miss any factors.

Worked Example 3: Find the GCF of \(20\) and \(30\)

Use factor pairs or list factors.

Factors of \(20\):

Factors of \(30\):

Common factors:

The greatest common factor is \(10\).

Finding the GCF of 3 numbers

You can also find the GCF of more than two numbers. The idea is the same: find the factors all the numbers share, then choose the greatest one.

Worked Example 4: Find the GCF of \(8\), \(12\), and \(20\)

List the factors of each number.

Factors of \(8\):

Factors of \(12\):

Factors of \(20\):

Now look for factors in all three lists:

The greatest common factor is \(4\).

How GCF helps with fractions

We can use the GCF to simplify fractions. To simplify a fraction, divide the numerator and denominator by their greatest common factor.

For example, simplify \(\frac{12}{18}\).

The GCF of \(12\) and \(18\) is \(6\).

Now divide both by \(6\):

So the simplified fraction is \(\frac{2}{3}\).

Tips for finding the GCF

• Always start by finding all the factors carefully.
• Check which factors are in both or all lists.
• Choose the greatest one, not just any common factor.
• If the only common factor is \(1\), then the GCF is \(1\).

Common mistake to avoid

Sometimes students find a common factor, but not the greatest one.

For example, for \(16\) and \(24\), the common factors are:

The GCF is \(8\), not \(4\), because \(8\) is bigger and still divides both numbers evenly.

Quick check

1. What is the GCF of \(6\) and \(9\)?
2. What is the GCF of \(15\) and \(25\)?
3. What is the GCF of \(9\), \(18\), and \(27\)?

Answers

1. Factors of \(6\): \(1, 2, 3, 6\); factors of \(9\): \(1, 3, 9\). GCF = \(3\).
2. Factors of \(15\): \(1, 3, 5, 15\); factors of \(25\): \(1, 5, 25\). GCF = \(5\).
3. Common factors of \(9\), \(18\), and \(27\) are \(1, 3, 9\). GCF = \(9\).

Summary

The greatest common factor is the biggest number that divides two or more numbers evenly.

To find it, list the factors of each number, find the common factors, and choose the greatest one.

GCF is useful for simplifying fractions and understanding number relationships.

Least Common Multiple (LCM) helps us find the smallest number that two or more numbers can all divide into evenly.

This is useful when we want to compare groups, line up repeating events, or find common denominators for fractions later on.

To understand LCM, we first need to remember what a multiple is.

A multiple of a number is what you get when you multiply that number by 1, 2, 3, 4, and so on.

For example, the multiples of 4 are:

\(4, 8, 12, 16, 20, 24, \dots\)

The multiples of 6 are:

\(6, 12, 18, 24, 30, \dots\)

A common multiple is a number that appears in both lists.

For 4 and 6, the common multiples are \(12, 24, 36, \dots\)

The least common multiple, or LCM, is the smallest common multiple.

So the LCM of 4 and 6 is \(12\).

We can write this as:

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

Why do we use the LCM?

The LCM helps when things happen in patterns or cycles.

For example, if one bell rings every 4 minutes and another bell rings every 6 minutes, the LCM tells us when they will ring together again.

Since \(\text{LCM}(4,6)=12\), both bells will ring together after 12 minutes.

How to find the LCM

There are two simple ways 5th Grade students can use.

1. List the multiples of each number until you find the first one they share.
2. Use skip counting to look for the first number both lists reach.

Let’s look at both ideas in action.

Method 1: List the multiples

Write the multiples of each number.

Then circle or notice the first number that appears in both lists.

Worked Example 1

Find \(\text{LCM}(3,5)\).

Multiples of 3:

\(3, 6, 9, 12, 15, 18, \dots\)

Multiples of 5:

\(5, 10, 15, 20, 25, \dots\)

The first number in both lists is \(15\).

So:

$$\text{LCM}(3,5)=15$$

Worked Example 2

Find \(\text{LCM}(4,10)\).

Multiples of 4:

\(4, 8, 12, 16, 20, 24, \dots\)

Multiples of 10:

\(10, 20, 30, 40, \dots\)

The first common multiple is \(20\).

So:

$$\text{LCM}(4,10)=20$$

Method 2: Use skip counting

Skip count by each number and listen for the first number that matches.

For example, for 2 and 7:

• By 2: \(2, 4, 6, 8, 10, 12, 14, \dots\)
• By 7: \(7, 14, 21, \dots\)

The first match is \(14\), so the LCM is \(14\).

Worked Example 3

Find \(\text{LCM}(6,8)\).

Multiples of 6:

\(6, 12, 18, 24, 30, 36, 42, 48, \dots\)

Multiples of 8:

\(8, 16, 24, 32, 40, 48, \dots\)

The first common multiple is \(24\).

So:

$$\text{LCM}(6,8)=24$$

Finding the LCM of 3 numbers

You can also find the LCM of three numbers by listing multiples and finding the smallest number all three share.

Worked Example 4

Find \(\text{LCM}(2,3,4)\).

Multiples of 2:

\(2, 4, 6, 8, 10, 12, 14, \dots\)

Multiples of 3:

\(3, 6, 9, 12, 15, \dots\)

Multiples of 4:

\(4, 8, 12, 16, 20, \dots\)

The first number in all three lists is \(12\).

So:

$$\text{LCM}(2,3,4)=12$$

Helpful tips

• The LCM must be a multiple of each number.
• The LCM is the smallest shared multiple, not just any shared multiple.
• If one number is already a multiple of the other, the bigger number is the LCM.

Example: for 5 and 10, since 10 is a multiple of 5, the LCM is \(10\).

Common mistakes to avoid

• Do not confuse factors and multiples. Factors go into a number. Multiples are made by multiplying.
• Do not stop too early. Keep listing until you find a number in both lists.
• Choose the least one. Even if there are many common multiples, the LCM is the smallest one.

Real-life example

A red light flashes every 3 seconds. A blue light flashes every 4 seconds.

When will they flash together?

Multiples of 3: \(3, 6, 9, 12, \dots\)

Multiples of 4: \(4, 8, 12, 16, \dots\)

The first time they match is \(12\).

So the lights will flash together every 12 seconds.

Let’s remember

• A multiple is the result of multiplying a number by 1, 2, 3, and so on.
• A common multiple is shared by two or more numbers.
• The least common multiple is the smallest shared multiple.
• You can find the LCM by listing multiples or skip counting.

When you see two or more numbers and need the smallest number they can all go into evenly, you are looking for the LCM.