Number Theory: Factors, Multiples, and Primes

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

For example, instead of calculating \(468 \div 3\), we can use a divisibility rule to quickly decide whether 468 is divisible by 3. This saves time and helps when finding factors, common factors, and prime factorizations.

In this lesson, you will learn how to test if a number is divisible by 2, 3, 4, 5, 6, 8, 9, and 10 by looking for patterns in its digits.

What does divisible mean?

A number is divisible by another number if it can be divided evenly, with no remainder.

For example:

• \(12\) is divisible by \(3\) because \(12 \div 3 = 4\).
• \(15\) is not divisible by \(4\) because \(15 \div 4\) leaves a remainder.

Why do divisibility rules work?

Our number system is based on tens. That means the ones digit, tens digit, hundreds digit, and so on create patterns. These patterns let us test divisibility by checking only certain digits or by adding digits together.

Divisibility Rules 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 two digits. If those two digits form a number 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 8: Look at the last three digits. If those three digits form a number divisible by \(8\), then the whole number is divisible by \(8\).
• 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 more closely at each rule.

1. Rule for 2

Even numbers are divisible by \(2\). So if the last digit is \(0, 2, 4, 6,\) or \(8\), the number is divisible by \(2\).

• \(34\) ends in \(4\), so it is divisible by \(2\).
• \(117\) ends in \(7\), so it is not divisible by \(2\).

2. Rule for 3

Add all the digits. If that total can be divided by \(3\), then the original number can too.

• For \(123\): \(1+2+3=6\). Since \(6\) is divisible by \(3\), \(123\) is divisible by \(3\).
• For \(124\): \(1+2+4=7\). Since \(7\) is not divisible by \(3\), \(124\) is not divisible by \(3\).

3. Rule for 4

Only the last two digits matter. If the last two digits make a number divisible by \(4\), then the whole number is divisible by \(4\).

• \(316\): last two digits are \(16\). Since \(16 \div 4 = 4\), \(316\) is divisible by \(4\).
• \(542\): last two digits are \(42\). Since \(42\) is not divisible by \(4\), \(542\) is not divisible by \(4\).

4. Rule for 5

If a number ends in \(0\) or \(5\), it is divisible by \(5\).

• \(90\) is divisible by \(5\).
• \(135\) is divisible by \(5\).
• \(132\) is not divisible by \(5\).

5. Rule for 6

To be divisible by \(6\), a number must pass two tests:

1. It must be divisible by \(2\).
2. It must be divisible by \(3\).

If both are true, then it is divisible by \(6\).

• \(54\): ends in \(4\), so it is divisible by \(2\). Also, \(5+4=9\), and \(9\) is divisible by \(3\). So \(54\) is divisible by \(6\).
• \(52\): ends in \(2\), so it is divisible by \(2\). But \(5+2=7\), and \(7\) is not divisible by \(3\). So \(52\) is not divisible by \(6\).

6. Rule for 8

Look at the last three digits. If that number is divisible by \(8\), then the whole number is divisible by \(8\).

• \(1{,}024\): last three digits are \(024\), which is \(24\). Since \(24 \div 8 = 3\), \(1{,}024\) is divisible by \(8\).
• \(2{,}138\): last three digits are \(138\). Since \(138\) is not divisible by \(8\), \(2{,}138\) is not divisible by \(8\).

7. Rule for 9

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

• \(729\): \(7+2+9=18\). Since \(18\) is divisible by \(9\), \(729\) is divisible by \(9\).
• \(731\): \(7+3+1=11\). Since \(11\) is not divisible by \(9\), \(731\) is not divisible by \(9\).

8. Rule for 10

If a number ends in \(0\), it is divisible by \(10\).

• \(150\) is divisible by \(10\).
• \(157\) is not divisible by \(10\).

Helpful Patterns to Notice

• Rules for 2, 5, and 10 depend only on the last digit.
• The rule for 4 depends on the last two digits.
• The rule for 8 depends on the last three digits.
• The rules for 3 and 9 use the sum of the digits.
• The rule for 6 combines the rules for 2 and 3.

These patterns make it easier to remember the rules.

Worked Example 1

Is \(246\) divisible by \(2\), \(3\), and \(6\)?

Step 1: Test for 2.

The number ends in \(6\), so it is divisible by \(2\).

Step 2: Test for 3.

Add the digits:

$$2+4+6=12$$

Since \(12\) is divisible by \(3\), \(246\) is divisible by \(3\).

Step 3: Test for 6.

A number is divisible by \(6\) if it is divisible by both \(2\) and \(3\). Since \(246\) passed both tests, it is divisible by \(6\).

Answer: \(246\) is divisible by \(2\), \(3\), and \(6\).

Worked Example 2

Is \(1{,}532\) divisible by \(4\), \(8\), and \(10\)?

Step 1: Test for 4.

Look at the last two digits: \(32\).

Since \(32 \div 4 = 8\), \(1{,}532\) is divisible by \(4\).

Step 2: Test for 8.

Look at the last three digits: \(532\).

Since \(532 \div 8 = 66\) remainder \(4\), \(532\) is not divisible by \(8\).

So \(1{,}532\) is not divisible by \(8\).

Step 3: Test for 10.

The number ends in \(2\), not \(0\), so it is not divisible by \(10\).

Answer: \(1{,}532\) is divisible by \(4\), but not by \(8\) or \(10\).

Worked Example 3

Which of these numbers are divisible by \(9\): \(351\), \(462\), \(745\)?

Test \(351\):

$$3+5+1=9$$

Since \(9\) is divisible by \(9\), \(351\) is divisible by \(9\).

Test \(462\):

$$4+6+2=12$$

Since \(12\) is not divisible by \(9\), \(462\) is not divisible by \(9\).

Test \(745\):

$$7+4+5=16$$

Since \(16\) is not divisible by \(9\), \(745\) is not divisible by \(9\).

Answer: Only \(351\) is divisible by \(9\).

Worked Example 4

Is \(4{,}560\) divisible by \(2, 3, 4, 5, 6, 8, 9,\) and \(10\)?

Let’s test each rule.

• By 2: It ends in \(0\), so yes.
• By 3: \(4+5+6+0=15\). Since \(15\) is divisible by \(3\), yes.
• By 4: Last two digits are \(60\). Since \(60\) is divisible by \(4\), yes.
• By 5: It ends in \(0\), so yes.
• By 6: It is divisible by both \(2\) and \(3\), so yes.
• By 8: Last three digits are \(560\). Since \(560 \div 8 = 70\), yes.
• By 9: The digit sum is \(15\), and \(15\) is not divisible by \(9\), so no.
• By 10: It ends in \(0\), so yes.

Answer: \(4{,}560\) is divisible by \(2, 3, 4, 5, 6, 8,\) and \(10\), but not by \(9\).

Common Mistakes to Avoid

• Mixing up the rules for 3 and 9. Both use digit sums, but the sum must be divisible by the number you are testing.
• Forgetting that 6 needs two tests. A number must be divisible by both \(2\) and \(3\).
• Checking all the digits for 4 or 8. For \(4\), only the last two digits matter. For \(8\), only the last three digits matter.
• Thinking every number ending in an even digit is divisible by 6. It must also pass the rule for \(3\).

Quick Check Tips

When you see a number, ask:

1. What is the last digit?
2. What are the last two or three digits?
3. What is the sum of the digits?

These three questions help you test many divisibility rules very quickly.

Summary

Divisibility rules are shortcuts that help you decide whether a number can be divided evenly by another number. They are especially useful when finding factors and solving number theory problems.

Remember these key ideas:

• 2: ends in \(0,2,4,6,8\)
• 3: digit sum is divisible by \(3\)
• 4: last two digits are divisible by \(4\)
• 5: ends in \(0\) or \(5\)
• 6: divisible by both \(2\) and \(3\)
• 8: last three digits are divisible by \(8\)
• 9: digit sum is divisible by \(9\)
• 10: ends in \(0\)

With practice, you will be able to use these patterns mentally and much faster than doing full division every time.

Prime and Composite Numbers

Numbers can tell us a lot about themselves by the factors they have. In this lesson, you will learn how to decide whether a whole number is prime, composite, or a special case like 1.

This is an important idea in number theory because factors help us understand how numbers are built. When we know whether a number is prime or composite, we can better solve problems with factors, multiples, and prime factorization.

First, what is a factor?

A factor of a number is a whole number that divides it evenly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these divides 12 exactly.

Prime numbers are whole numbers greater than 1 that have exactly 2 factors: 1 and the number itself.

Examples of prime numbers are 2, 3, 5, 7, and 11.

For example, 7 is prime because its only factors are 1 and 7.

Composite numbers are whole numbers greater than 1 that have more than 2 factors.

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

For example, 8 is composite because it has factors 1, 2, 4, and 8.

The number 1 is special.

The number 1 has only 1 factor, which is 1 itself. A prime number must have exactly 2 factors, and a composite number must have more than 2 factors. Since 1 has neither of these, it is neither prime nor composite.

This is why we say:

• Prime numbers have exactly 2 factors.
• Composite numbers have more than 2 factors.
• The number 1 is neither prime nor composite.

How to tell if a number is prime or composite

1. Make sure the number is a whole number greater than 1.
2. Find its factors.
3. Count how many factors it has.
4. If it has exactly 2 factors, it is prime.
5. If it has more than 2 factors, it is composite.

You can also think about multiplication. If a number can be made by multiplying two smaller whole numbers, then it is composite.

For example, since $$12 = 3 \times 4$$, the number 12 is composite. But 13 cannot be written as a product of two smaller whole numbers, so 13 is prime.

A helpful fact about 2

The number 2 is the smallest prime number. It is also the only even prime number.

Why? Every other even number is divisible by 2, so it has at least the factors 1, 2, and itself. That means every even number greater than 2 is composite.

For example:

• 2 has factors 1 and 2, so it is prime.
• 4 has factors 1, 2, and 4, so it is composite.
• 6 has factors 1, 2, 3, and 6, so it is composite.

Worked Example 1: Is 5 prime or composite?

Find the factors of 5.

• 1 divides 5 evenly.
• 5 divides 5 evenly.

The factors of 5 are 1 and 5.

Since 5 has exactly 2 factors, 5 is prime.

Worked Example 2: Is 9 prime or composite?

Find the factors of 9.

• 1 divides 9 evenly.
• 3 divides 9 evenly because $$9 = 3 \times 3$$.
• 9 divides 9 evenly.

The factors of 9 are 1, 3, and 9.

Since 9 has more than 2 factors, 9 is composite.

Worked Example 3: What about 1?

The factors of 1 are just 1.

So 1 has only 1 factor.

A prime number must have exactly 2 factors. A composite number must have more than 2 factors. Therefore, 1 is neither prime nor composite.

Worked Example 4: Classify 17

Let us check its factors.

17 is divisible evenly by 1 and 17.

It is not divisible evenly by 2, 3, 4, 5, 6, 7, 8, or 9.

So the only factors of 17 are 1 and 17.

Since it has exactly 2 factors, 17 is prime.

Prime and composite numbers from 1 to 20

It can help to see a list.

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

Common mistakes to avoid

• Do not say 1 is prime. It is neither prime nor composite.
• Do not forget that a prime number must be greater than 1.
• Do not think all odd numbers are prime. For example, 9 and 15 are odd, but they are composite.
• Remember that 2 is even, but it is still prime.

Quick check strategy

When checking a number, ask:

1. Is the number 1? If yes, it is neither.
2. Can it be divided evenly by a number other than 1 and itself?
3. If no, it is prime.
4. If yes, it is composite.

For example, to test 15, ask whether it can be divided by a number other than 1 and 15. Yes, because $$15 = 3 \times 5$$. So 15 is composite.

Summary

Prime and composite numbers are classified by their factors. A prime number is a whole number greater than 1 with exactly 2 factors: 1 and itself.

A composite number is a whole number greater than 1 with more than 2 factors. The number 1 is special because it has only 1 factor, so it is neither prime nor composite.

Whenever you are unsure, list the factors of the number and count them. That will help you classify the number correctly.

Prime Factorization means writing a number as a product of prime numbers.

Before we learn how to do that, let’s remember two important words:

• Prime number: a number greater than 1 that has exactly 2 factors: 1 and itself. Examples: 2, 3, 5, 7, 11.
• Composite number: a number greater than 1 that has more than 2 factors. Examples: 4, 6, 8, 9, 10.

Prime factorization is useful because it shows the “building blocks” of a number. Every composite number can be broken apart into primes.

For example, the number 12 can be written as:

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

Since 2 and 3 are both prime, this is the prime factorization of 12.

This lesson will show you how to:

• find prime factorization using a factor tree,
• find prime factorization using the ladder method,
• check that all factors are prime,
• use exponents to write repeated prime factors.

Main Idea: Keep breaking a number into factors until every factor is prime.

It does not matter which pair of factors you choose first. If you keep going until all the factors are prime, you will always end with the same prime factorization.

Method 1: Factor Tree

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

Here are the steps:

1. Write the number.
2. Choose any two factors of the number.
3. If a factor is composite, break it into more factors.
4. Circle or list the prime numbers at the ends.
5. Multiply the prime numbers together.

Method 2: Ladder Method

In the ladder method, you divide the number by prime numbers one step at a time.

Here are the steps:

1. Write the number.
2. Divide by the smallest prime number that goes into it evenly.
3. Write the quotient.
4. Keep dividing by prime numbers until the quotient is 1.
5. The prime numbers you used are the prime factors.

Important Reminder: 1 is not a prime number, so it is never included in prime factorization.

Worked Example 1: Prime Factorization of 18 using a Factor Tree

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

$$18 = 2 \times 9$$

The number 2 is prime, but 9 is composite. So break 9 into 3 and 3.

$$9 = 3 \times 3$$

Now all the factors are prime:

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

Using an exponent, we can write this as:

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

This means there is one 2 and two 3s in the prime factorization.

Worked Example 2: Prime Factorization of 24 using a Factor Tree

Start with 24. One possible split is 4 and 6.

$$24 = 4 \times 6$$

Both 4 and 6 are composite, so keep breaking them apart.

$$4 = 2 \times 2$$

$$6 = 2 \times 3$$

Now all the factors are prime, so:

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

There are three 2s, so we can use an exponent:

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

Notice that you could have started with a different factor pair, like 3 and 8. You would still end with the same prime factors.

Worked Example 3: Prime Factorization of 36 using the Ladder Method

We divide by prime numbers step by step.

Start with 36.

36 is even, so divide by 2:

$$36 \div 2 = 18$$

18 is also even, so divide by 2 again:

$$18 \div 2 = 9$$

9 is not divisible by 2, but it is divisible by 3:

$$9 \div 3 = 3$$

Divide by 3 again:

$$3 \div 3 = 1$$

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

So the prime factorization is:

$$36 = 2 \times 2 \times 3 \times 3$$

Using exponents:

$$36 = 2^2 \times 3^2$$

Worked Example 4: Prime Factorization of 60 using the Ladder Method

Let’s divide by the smallest prime numbers first.

$$60 \div 2 = 30$$

$$30 \div 2 = 15$$

$$15 \div 3 = 5$$

$$5 \div 5 = 1$$

The prime factors are 2, 2, 3, and 5.

So:

$$60 = 2 \times 2 \times 3 \times 5$$

Using an exponent for the repeated 2:

$$60 = 2^2 \times 3 \times 5$$

How to Know When You Are Finished

• Every factor left must be a prime number.
• If you still have a composite number, keep factoring.
• Do not stop too early.

For example, if you write:

$$20 = 4 \times 5$$

this is not prime factorization yet, because 4 is composite.

You must keep going:

$$4 = 2 \times 2$$

So the prime factorization is:

$$20 = 2 \times 2 \times 5 = 2^2 \times 5$$

Using Exponents

An exponent is a short way to show repeated multiplication of the same number.

For example:

• $$2 \times 2 = 2^2$$
• $$3 \times 3 \times 3 = 3^3$$
• $$5 \times 5 \times 5 \times 5 = 5^4$$

So if a prime factor appears more than once, you can use an exponent to write it more neatly.

Why Prime Factorization Matters

Prime factorization helps with many number problems. It is especially useful when finding:

• common factors,
• greatest common factor (GCF),
• common multiples,
• least common multiple (LCM).

When you know the prime factors of numbers, it becomes easier to compare them.

Common Mistakes to Avoid

• Including 1 as a prime factor — 1 is not prime.
• Stopping too soon — make sure all factors are prime.
• Using a composite number in the final answer — the final answer must use only primes.
• Forgetting repeated factors — count how many times each prime appears.

Quick Check

Find the prime factorization of 45.

One way is:

$$45 = 5 \times 9$$

$$9 = 3 \times 3$$

So:

$$45 = 3 \times 3 \times 5 = 3^2 \times 5$$

Another Quick Check

Find the prime factorization of 72.

Using a factor tree or ladder method:

$$72 = 2 \times 36$$

$$36 = 2 \times 18$$

$$18 = 2 \times 9$$

$$9 = 3 \times 3$$

So:

$$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$$

Summary

Prime factorization means writing a composite number as a product of prime numbers only.

You can find prime factorization by using a factor tree or the ladder method.

Keep factoring until every number is prime. If a prime factor repeats, you can use an exponent to write it in a shorter way.

For example:

$$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$$

Once you understand prime factorization, you will be ready to use it to solve bigger problems with factors and multiples.

Greatest Common Factor (GCF) is the biggest number that can divide two or more numbers with no remainder.

To understand GCF, we first need to remember what a factor is. 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:

$$1, 2, 3, 4, 6, 12$$

This is because:

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

When two or more numbers share some of the same factors, those are called common factors.

The greatest common factor is the largest factor they have in common.

For example, for \(12\) and \(18\):

• Factors of \(12\): \(1, 2, 3, 4, 6, 12\)
• Factors of \(18\): \(1, 2, 3, 6, 9, 18\)

The common factors are \(1, 2, 3, 6\). The greatest of these is \(6\).

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

$$\text{GCF}(12,18)=6$$

There are two main ways to find the GCF:

• Listing factors
• Using prime factorization

Let’s learn both methods.

Method 1: Listing Factors

In this method, list all the factors of each number. Then circle or pick out the factors they share. The greatest shared factor is the GCF.

Worked Example 1: Find the GCF of \(16\) and \(24\)

Step 1: List the factors.

• Factors of \(16\): \(1, 2, 4, 8, 16\)
• Factors of \(24\): \(1, 2, 3, 4, 6, 8, 12, 24\)

Step 2: Find the common factors.

The common factors are \(1, 2, 4, 8\).

Step 3: Choose the greatest one.

The greatest common factor is \(8\).

$$\text{GCF}(16,24)=8$$

This method works well for smaller numbers because the factor lists are not too long.

Method 2: Prime Factorization

A prime number is a number greater than \(1\) that has exactly two factors: \(1\) and itself. Examples are \(2, 3, 5, 7, 11\).

Prime factorization means writing a number as a product of prime numbers.

For example:

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

or using exponents:

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

To find the GCF using prime factorization:

1. Write each number as a product of prime factors.
2. Find the prime factors that appear in every number.
3. Multiply the common prime factors together.

Worked Example 2: Find the GCF of \(18\) and \(30\)

Step 1: Prime factorize each number.

$$18 = 2 \times 3 \times 3 = 2 \times 3^2$$ $$30 = 2 \times 3 \times 5$$

Step 2: Find the prime factors they have in common.

Both numbers have one \(2\) and one \(3\).

Step 3: Multiply the common prime factors.

$$2 \times 3 = 6$$

So:

$$\text{GCF}(18,30)=6$$

Notice that even though \(18\) has two \(3\)'s, \(30\) has only one \(3\). So we only use the factor \(3\) once.

Important Idea: When using prime factorization, use only the prime factors that all the numbers share, and only as many times as they appear in every number.

Worked Example 3: Find the GCF of \(24\) and \(36\)

Step 1: Prime factorize each number.

$$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$$ $$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$$

Step 2: Find the common prime factors.

• Both have two \(2\)'s.
• Both have one \(3\).

Step 3: Multiply them.

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

So:

$$\text{GCF}(24,36)=12$$

This example shows that we take the factors shared by both numbers. We do not take the extra \(2\) from \(24\) or the extra \(3\) from \(36\), because those are not shared.

Finding the GCF of More Than Two Numbers

You can also find the GCF of three or more numbers. The idea is the same: find the largest factor that all the numbers share.

Worked Example 4: Find the GCF of \(12\), \(18\), and \(30\)

Step 1: Prime factorize each number.

$$12 = 2^2 \times 3$$ $$18 = 2 \times 3^2$$ $$30 = 2 \times 3 \times 5$$

Step 2: Look for prime factors that appear in all three numbers.

• All three have one \(2\).
• All three have one \(3\).

Step 3: Multiply the common prime factors.

$$2 \times 3 = 6$$

So:

$$\text{GCF}(12,18,30)=6$$

How to Check Your Answer

After you find a GCF, ask yourself two questions:

• Does this number divide evenly into all the given numbers?
• Is there any larger number that also divides evenly into all of them?

If your answer divides all the numbers and no larger common factor exists, then your GCF is correct.

Why GCF Matters

The GCF is useful when you want to:

• Split items into the largest equal groups
• Simplify some number problems
• Understand how numbers are built from factors

For example, if you have \(20\) apples and \(28\) oranges and want to make the greatest number of equal fruit bags with no fruit left over, you need the GCF of \(20\) and \(28\).

Since:

• \(20 = 2^2 \times 5\)
• \(28 = 2^2 \times 7\)

The common prime factors are \(2^2\), so:

$$\text{GCF}(20,28)=4$$

That means you can make \(4\) equal bags.

Common Mistakes to Avoid

• Mixing up factors and multiples: Factors divide a number evenly. Multiples are found by multiplying a number.
• Choosing any common factor instead of the greatest one: The answer must be the largest shared factor.
• Using prime factors that are not shared by all numbers: Only use the factors every number has in common.
• Using a prime factor too many times: If one number has \(2^3\) and another has \(2^2\), you can only use \(2^2\).

Quick Steps to Remember

If the numbers are small:

1. List the factors of each number.
2. Find the common factors.
3. Choose the greatest one.

If the numbers are larger:

1. Write the prime factorization of each number.
2. Find the prime factors they all share.
3. Multiply those shared prime factors.

Summary

The greatest common factor (GCF) is the largest factor shared by two or more numbers.

You can find it by listing factors or by using prime factorization.

When using prime factorization, choose only the prime factors that appear in every number, and only the number of times they are shared.

Learning GCF helps you solve problems with equal groups and understand number relationships better.

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

For example, if you want to know when two events happen at the same time again, the LCM can help. If one event happens every 4 minutes and another happens every 6 minutes, the LCM tells you the first time they will match up again.

In this lesson, you will learn what the LCM means, how to find it by listing multiples, and how to find it by prime factorization.

What is a multiple?

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

• Multiples of 3: \(3, 6, 9, 12, 15, 18, \dots\)
• Multiples of 5: \(5, 10, 15, 20, 25, 30, \dots\)

A common multiple is a number that appears in the multiples of both numbers.

For 3 and 5, some common multiples are \(15, 30, 45, \dots\)

The least common multiple is the smallest nonzero common multiple.

So for 3 and 5, the LCM is \(15\).

Method 1: Find the LCM by listing multiples

This method works well when the numbers are small.

1. Write the first several multiples of each number.
2. Look for the first multiple that appears in every list.
3. That number is the LCM.

Worked Example 1: Find the LCM of 4 and 6

List the multiples:

• Multiples of 4: \(4, 8, 12, 16, 20, 24, \dots\)
• Multiples of 6: \(6, 12, 18, 24, 30, \dots\)

The first number that appears in both lists is \(12\).

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

Worked Example 2: Find the LCM of 8 and 10

List the multiples:

• Multiples of 8: \(8, 16, 24, 32, 40, 48, \dots\)
• Multiples of 10: \(10, 20, 30, 40, 50, \dots\)

The first common multiple is \(40\).

So, $$\text{LCM}(8,10)=40$$

Method 2: Find the LCM by prime factorization

This method is very useful for larger numbers or for finding the LCM of more than two numbers.

Step 1: Write each number as a product of prime numbers.

Step 2: For each prime number, use the greatest number of times it appears in any factorization.

Step 3: Multiply those prime factors together.

Worked Example 3: Find the LCM of 12 and 18

Prime factorize each number:

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

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

Now choose the greatest power of each prime:

• For 2, the greatest power is \(2^2\)
• For 3, the greatest power is \(3^2\)

Multiply them:

$$\text{LCM}(12,18)=2^2\times 3^2=4\times 9=36$$

So the LCM is \(36\).

Worked Example 4: Find the LCM of 6, 8, and 15

Prime factorize each number:

$$6=2\times 3$$

$$8=2^3$$

$$15=3\times 5$$

Now choose the greatest power of each prime that appears:

• For 2, use \(2^3\)
• For 3, use \(3\)
• For 5, use \(5\)

Multiply them:

$$\text{LCM}(6,8,15)=2^3\times 3\times 5=8\times 3\times 5=120$$

So the LCM is \(120\).

How do you know your answer is correct?

The LCM must be a multiple of each number.

• For \(\text{LCM}(4,6)=12\): 12 is divisible by 4 and by 6.
• For \(\text{LCM}(12,18)=36\): 36 is divisible by 12 and by 18.

Helpful tips

• The LCM is never smaller than the numbers you started with.
• If one number is already a multiple of the other, the larger number is the LCM.
• Listing multiples is good for small numbers.
• Prime factorization is usually faster for larger numbers.

Common mistake to avoid

Do not stop at just any common multiple. You need the least common multiple, which means the smallest one greater than 0.

For example, 24 is a common multiple of 4 and 6, but it is not the LCM because 12 is smaller and is also a common multiple.

Summary

The least common multiple is the smallest nonzero number that is a multiple of all the given numbers.

You can find the LCM by listing multiples or by using prime factorization.

When using prime factorization, include each prime factor the greatest number of times it appears in any one number, then multiply.

With practice, you can use the LCM to solve number problems and real-life timing problems.

Applying GCF with the Distributive Property

Sometimes in math, we can take a sum like \(24 + 36\) and rewrite it in a different way. This helps us see how the numbers are connected.

One useful tool is the greatest common factor, or GCF. Another useful tool is the distributive property. When we use them together, we can rewrite expressions in a simpler and more meaningful form.

For example, instead of writing \(24 + 36\), we can write:

$$24 + 36 = 12(2 + 3)$$

This works because \(12\) is the greatest common factor of 24 and 36.

In this lesson, you will learn how to:

• find the GCF of two or more numbers,
• factor the GCF out of an expression,
• use the distributive property to check your work.

1. Review: What is the GCF?

The greatest common factor of two or more numbers is the largest whole number that divides evenly into all of them.

For example, the factors of 24 are:

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

The factors of 36 are:

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

The common factors are \(1, 2, 3, 4, 6, 12\). The greatest one is \(12\), so:

$$\text{GCF of } 24 \text{ and } 36 = 12$$

2. Review: What is the distributive property?

The distributive property says that multiplying a number outside parentheses gives the same result as multiplying it by each number inside.

For example:

$$3(4 + 5) = 3 \cdot 4 + 3 \cdot 5 = 12 + 15 = 27$$

We can also think about this idea backward. If two numbers in a sum both have a common factor, we can factor out that common factor and write the sum with parentheses.

For example:

$$12 + 18 = 6(2 + 3)$$

This is true because:

$$6(2 + 3) = 6 \cdot 2 + 6 \cdot 3 = 12 + 18$$

3. What does it mean to factor out the GCF?

Factoring out the GCF means finding the greatest number that divides each term, then rewriting the expression as a product.

Here is the basic idea:

$$a + b = \text{GCF}(a,b)(\text{new first number} + \text{new second number})$$

To do this:

1. Find the GCF of the numbers.
2. Divide each number by the GCF.
3. Put those answers inside parentheses.
4. Check by distributing.

4. Worked Example 1

Rewrite \(18 + 30\) by factoring out the GCF.

Step 1: Find the GCF.

Factors of 18: \(1, 2, 3, 6, 9, 18\)

Factors of 30: \(1, 2, 3, 5, 6, 10, 15, 30\)

The greatest common factor is \(6\).

Step 2: Divide each term by 6.

$$18 \div 6 = 3$$ $$30 \div 6 = 5$$

Step 3: Write the factored form.

$$18 + 30 = 6(3 + 5)$$

Step 4: Check by distributing.

$$6(3 + 5) = 6 \cdot 3 + 6 \cdot 5 = 18 + 30$$

So the correct rewrite is:

$$\boxed{6(3 + 5)}$$

5. Worked Example 2

Rewrite \(20 + 45\) by factoring out the GCF.

Step 1: Find the GCF.

Factors of 20: \(1, 2, 4, 5, 10, 20\)

Factors of 45: \(1, 3, 5, 9, 15, 45\)

The greatest common factor is \(5\).

Step 2: Divide each term by 5.

$$20 \div 5 = 4$$ $$45 \div 5 = 9$$

Step 3: Write the factored form.

$$20 + 45 = 5(4 + 9)$$

Step 4: Check.

$$5(4 + 9) = 5 \cdot 4 + 5 \cdot 9 = 20 + 45$$

6. Worked Example 3

Rewrite \(14 + 21 + 35\) by factoring out the GCF.

This time there are three terms. We still use the same idea.

Step 1: Find the GCF of all three numbers.

Factors of 14: \(1, 2, 7, 14\)

Factors of 21: \(1, 3, 7, 21\)

Factors of 35: \(1, 5, 7, 35\)

The greatest common factor of all three is \(7\).

Step 2: Divide each term by 7.

$$14 \div 7 = 2$$ $$21 \div 7 = 3$$ $$35 \div 7 = 5$$

Step 3: Write the expression.

$$14 + 21 + 35 = 7(2 + 3 + 5)$$

Step 4: Check.

$$7(2 + 3 + 5) = 7 \cdot 2 + 7 \cdot 3 + 7 \cdot 5 = 14 + 21 + 35$$

7. Worked Example 4

Rewrite \(16 + 24\) by factoring out the GCF.

Step 1: Find the GCF.

Factors of 16: \(1, 2, 4, 8, 16\)

Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)

The GCF is \(8\).

Step 2: Divide each term by 8.

$$16 \div 8 = 2$$ $$24 \div 8 = 3$$

Step 3: Write the factored form.

$$16 + 24 = 8(2 + 3)$$

Step 4: Check.

$$8(2 + 3) = 8 \cdot 2 + 8 \cdot 3 = 16 + 24$$

8. A helpful way to think about it

When you factor out the GCF, you are asking:

“What is the biggest equal group I can take from every term?”

For example, in \(24 + 36\), both numbers can be split into groups of 12:

$$24 = 12 \cdot 2$$ $$36 = 12 \cdot 3$$

So:

$$24 + 36 = 12 \cdot 2 + 12 \cdot 3 = 12(2 + 3)$$

This shows the structure of the expression more clearly.

9. Common mistakes to avoid

• Using a common factor that is not the greatest.
  You can write \(24 + 36 = 6(4 + 6)\), and that is true. But it is better to factor out the greatest common factor: \(12(2 + 3)\).
• Forgetting to divide every term.
  If you factor out 5 from \(20 + 45\), both 20 and 45 must be divided by 5.
• Not checking with the distributive property.
  Always multiply back to make sure your expression matches the original one.

10. Quick practice ideas

Try rewriting these by factoring out the GCF:

• \(12 + 28\)
• \(27 + 36\)
• \(15 + 25 + 35\)

You can solve each one by:

1. finding the GCF,
2. dividing each term by the GCF,
3. writing the new expression with parentheses.

11. Summary

To apply the GCF with the distributive property, first find the greatest common factor of all the terms. Then divide each term by that GCF and write the expression as a product with parentheses.

This helps you rewrite expressions such as:

$$24 + 36 = 12(2 + 3)$$

You can always check your answer by distributing to see if you get the original expression back.

Applying LCM to Cycles and Synchronicity

Sometimes things happen in repeating patterns. A bus may arrive every 10 minutes. A light may flash every 6 seconds. A bell may ring every 15 minutes.

When two or more repeating events start together, we may want to know when they will happen together again. To solve this, we use the LCM.

LCM means Least Common Multiple. It is the smallest positive number that is a multiple of each number.

In this lesson, you will learn how to use the LCM to solve problems about cycles, overlapping events, and synchronicity. Synchronicity means things happening at the same time.

Why does LCM help?

If one event repeats every 4 minutes, its times are multiples of 4:

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

If another event repeats every 6 minutes, its times are multiples of 6:

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

The first time both happen together again is at the smallest common multiple, which is \(12\).

So, the LCM tells us the first time the cycles match up again.

How to find the LCM

There are two helpful methods.

Method 1: List multiples

• Write the multiples of each number.
• Find the smallest number that appears in both lists.

Example: Find the LCM of \(8\) and \(12\).

• Multiples of \(8\): \(8, 16, 24, 32, \dots\)
• Multiples of \(12\): \(12, 24, 36, \dots\)

The first common multiple is \(24\), so:

$$\text{LCM}(8,12)=24$$

Method 2: Use prime factorization

Break each number into prime factors. Then use each prime factor the greatest number of times it appears.

Example: Find the LCM of \(12\) and \(18\).

Prime factorization:

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

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

Take the greatest power of each prime:

• For \(2\), use \(2^2\)
• For \(3\), use \(3^2\)

So:

$$\text{LCM}(12,18)=2^2 \times 3^2 = 4 \times 9 = 36$$

Using LCM in cycle problems

Cycle problems often ask questions like these:

• When will two lights flash together again?
• When will two runners be at the starting point together again?
• When will buses arrive together again?
• What is the least time before repeating schedules match?

These are all LCM problems because the events repeat at regular intervals.

Steps for solving cycle and synchronicity problems

1. Find the repeating times or intervals.
2. Find the LCM of those numbers.
3. Use the LCM to answer the question.
4. If needed, add the LCM to the starting time.

Worked Example 1: Two flashing lights

A red light flashes every 4 seconds. A blue light flashes every 6 seconds. If they flash together now, when will they flash together again?

Step 1: Identify the intervals.

• Red light: every \(4\) seconds
• Blue light: every \(6\) seconds

Step 2: Find the LCM of \(4\) and \(6\).

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

Multiples of \(6\): \(6, 12, 18, \dots\)

The least common multiple is \(12\).

Step 3: Answer the question.

They will flash together again in 12 seconds.

Worked Example 2: Overlapping schedules

One school club meets every 8 days. Another club meets every 12 days. If both clubs meet today, after how many days will they meet on the same day again?

Step 1: Identify the intervals.

• First club: every \(8\) days
• Second club: every \(12\) days

Step 2: Find the LCM.

Prime factorization:

$$8 = 2^3$$

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

Take the greatest power of each prime:

$$\text{LCM}(8,12)=2^3 \times 3 = 8 \times 3 = 24$$

Step 3: Answer the question.

The clubs will meet on the same day again after 24 days.

Worked Example 3: Finding the clock time

A small bell rings every 9 minutes. A large bell rings every 15 minutes. They both ring at 8:00 a.m. At what time will they next ring together?

Step 1: Identify the intervals.

• Small bell: every \(9\) minutes
• Large bell: every \(15\) minutes

Step 2: Find the LCM of \(9\) and \(15\).

Prime factorization:

$$9 = 3^2$$

$$15 = 3 \times 5$$

Take the greatest power of each prime:

$$\text{LCM}(9,15)=3^2 \times 5 = 9 \times 5 = 45$$

Step 3: Add the time.

They ring together every \(45\) minutes.

Starting at 8:00 a.m.:

$$8{:}00 + 45\text{ minutes} = 8{:}45\text{ a.m.}$$

Answer: They will next ring together at 8:45 a.m.

Worked Example 4: Three repeating events

Three machines beep every 3 minutes, 4 minutes, and 6 minutes. If they beep together now, after how many minutes will all three beep together again?

Step 1: Find the LCM of \(3\), \(4\), and \(6\).

Prime factorization:

$$3 = 3$$

$$4 = 2^2$$

$$6 = 2 \times 3$$

Take the greatest power of each prime:

• For \(2\), use \(2^2\)
• For \(3\), use \(3\)

So:

$$\text{LCM}(3,4,6)=2^2 \times 3 = 4 \times 3 = 12$$

Step 2: Answer the question.

All three machines will beep together again after 12 minutes.

How LCM connects to common denominators

LCM is also used with fractions. When fractions have different denominators, we often use the LCM of the denominators to find the least common denominator.

Example: For \(\frac{1}{4}\) and \(\frac{1}{6}\), the denominators are \(4\) and \(6\).

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

So the least common denominator is \(12\).

This is another way LCM helps when different repeating parts need to match.

Tips for solving word problems

• If events repeat at regular times, think about LCM.
• If the question asks when they happen together again, you probably need the LCM.
• If a starting clock time is given, first find the LCM, then add that amount of time.
• If there are three or more cycles, find the LCM of all the intervals.

Be careful!

• Do not confuse LCM with GCF. The GCF finds the greatest factor numbers share. The LCM finds the smallest multiple they share.
• Make sure the numbers you use are the repeat intervals, not the starting time.
• If the problem says the events start together now, the answer is usually the next time, not right now.

Quick check

• Every 5 minutes and every 10 minutes: \(\text{LCM}(5,10)=10\)
• Every 6 days and every 14 days: \(\text{LCM}(6,14)=42\)
• Every 2 hours, 3 hours, and 4 hours: \(\text{LCM}(2,3,4)=12\)

Summary

The Least Common Multiple helps us solve problems about repeating events. When cycles start together and repeat over and over, the LCM tells us the first time they will match again.

To solve these problems:

1. Find the repeat intervals.
2. Find their LCM.
3. Use that number as the time, day, or amount until the events happen together again.

LCM is useful for:

• flashing lights,
• bells and alarms,
• meeting schedules,
• repeating routes,
• and common denominators in fractions.

Whenever you see repeating events that need to line up, think: Find the LCM!