Number Theory and Exponents

Divisibility Algorithms are quick rules that help you decide whether one whole number can be divided evenly by another number without doing long division.

If a number is divisible by another number, it means the division has no remainder. For example, since \(24 \div 6 = 4\), we say that 24 is divisible by 6.

These rules are helpful when you need to work with large numbers, find factors, identify prime numbers, or break numbers into smaller parts.

In this lesson, you will learn simple divisibility tests for several single-digit numbers and how to use them step by step.

Why divisibility rules work

Every whole number is built from digits. Divisibility algorithms use patterns in those digits to tell us whether a number can be divided evenly.

For example, numbers ending in 0, 2, 4, 6, or 8 are even, so they are divisible by 2. Numbers ending in 0 or 5 are divisible by 5. Some rules use the last digit, while others use the sum of the digits.

Main divisibility rules

• Divisible by 2: The last digit is 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 that two-digit number is divisible by 4, then the whole number is divisible by 4.
• Divisible by 5: The last digit is 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 that three-digit number is 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 last digit is 0.

Important idea: Some divisibility tests are based on other tests. For example, divisibility by 6 depends on divisibility by 2 and 3.

How to use a divisibility algorithm

1. Choose the divisor you are testing, such as 2, 3, 4, 5, 6, 8, 9, or 10.
2. Use the matching divisibility rule.
3. Check whether the number fits the rule.
4. Decide whether the number is divisible or not divisible.

Worked Example 1: Testing divisibility by 2 and 5

Is \(3{,}470\) divisible by 2? Is it divisible by 5?

For 2, check the last digit. The last digit is 0, and 0 is an even digit. So \(3{,}470\) is divisible by 2.

For 5, check the last digit again. A number is divisible by 5 if it ends in 0 or 5. Since \(3{,}470\) ends in 0, it is divisible by 5.

We can write:

$$3{,}470 \text{ is divisible by } 2 \text{ and } 5.$$

Worked Example 2: Testing divisibility by 3 and 9

Is \(5{,}328\) divisible by 3? Is it divisible by 9?

Add the digits:

$$5 + 3 + 2 + 8 = 18$$

Since 18 is divisible by 3, the number \(5{,}328\) is divisible by 3.

Since 18 is also divisible by 9, the number \(5{,}328\) is divisible by 9 too.

So:

$$5{,}328 \div 3 \text{ has no remainder, and } 5{,}328 \div 9 \text{ has no remainder.}$$

Worked Example 3: Testing divisibility by 4 and 8

Is \(7{,}416\) divisible by 4? Is it divisible by 8?

For 4, look at the last two digits. The last two digits are 16.

Since \(16 \div 4 = 4\), 16 is divisible by 4. So \(7{,}416\) is divisible by 4.

For 8, look at the last three digits. The last three digits are 416.

Check whether 416 is divisible by 8:

$$416 \div 8 = 52$$

Since the division is exact, \(7{,}416\) is divisible by 8.

Worked Example 4: Testing divisibility by 6

Is \(2{,}154\) divisible by 6?

To be divisible by 6, a number must be divisible by both 2 and 3.

First, test divisibility by 2. The last digit is 4, so the number is divisible by 2.

Next, test divisibility by 3. Add the digits:

$$2 + 1 + 5 + 4 = 12$$

Since 12 is divisible by 3, \(2{,}154\) is divisible by 3.

Because \(2{,}154\) is divisible by both 2 and 3, it is divisible by 6.

Using divisibility rules to break apart numbers

Divisibility algorithms can help you decompose, or break apart, a number into factors.

Suppose you want to break apart \(180\).

• It ends in 0, so it is divisible by 10.
• Its last digit is 0, so it is divisible by 2 and 5.
• The digit sum is \(1+8+0=9\), so it is divisible by 3 and 9.
• The last two digits are 80, so it is divisible by 4.

This tells us that 180 has many factors. For example:

$$180 = 18 \times 10 = 36 \times 5 = 9 \times 20$$

Divisibility rules help you find these factor pairs much faster.

Common mistakes to avoid

• Mixing up the rules for 3 and 9: Both use the sum of the digits, but make sure you check whether the sum is divisible by 3 or by 9.
• Using only one part of the rule for 6: A number must pass both tests, for 2 and for 3.
• Looking at the whole number instead of the ending digits: For 4, check the last two digits. For 8, check the last three digits.
• Forgetting what divisible means: It means there is no remainder.

Quick check examples

• \(246\): divisible by 2 because it ends in 6.
• \(246\): divisible by 3 because \(2+4+6=12\), and 12 is divisible by 3.
• \(246\): not divisible by 9 because 12 is not divisible by 9.
• \(1{,}235\): divisible by 5 because it ends in 5.
• \(1{,}232\): divisible by 4 because the last two digits, 32, are divisible by 4.

Summary

Divisibility algorithms are fast methods for deciding whether a number can be divided evenly by another number.

You can often check just the last digit, the last two or three digits, or the sum of the digits. Learning these patterns makes it easier to find factors, simplify number problems, and work with larger numbers confidently.

Fundamental Theorem of Arithmetic is a big name for an important idea about whole numbers.

It says that every whole number greater than 1 is either:

• a prime number, or
• can be written as a product of prime numbers.

It also says this prime factorization is unique. That means there is only one correct list of prime factors for a number, except for the order they are written in.

For example, the number 12 can be written as:

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

You could also write it as:

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

These are not different factorizations, because they use the same prime factors. The order does not matter.

1. Review: Prime and Composite Numbers

To understand this theorem, you need to know the difference between prime and composite numbers.

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

Examples of prime numbers are:

• 2
• 3
• 5
• 7
• 11

A composite number is a whole number greater than 1 that has more than two factors.

Examples of composite numbers are:

• 4
• 6
• 8
• 9
• 12

The number 1 is neither prime nor composite.

2. What Does “Product of Prime Numbers” Mean?

A product is the answer to a multiplication problem. So a product of prime numbers means multiplying primes together.

For example:

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

Here, 18 is written as a product of prime numbers because 2 and 3 are both prime.

This is called the prime factorization of 18.

3. Statement of the Fundamental Theorem of Arithmetic

Here is the theorem in simple words:

Every integer greater than 1 is either prime or can be broken into prime factors in exactly one way, not counting order.

This theorem has two main parts:

1. Existence: Every number greater than 1 can be written using primes.
2. Uniqueness: The prime factorization is the same every time, except for the order.

This means prime numbers are like the building blocks of whole numbers.

4. How to Find Prime Factorization

One common way is to use a factor tree.

Start with the number. Break it into factors. If a factor is composite, keep breaking it apart. Stop when every branch ends in a prime number.

Then multiply the prime numbers together.

You can also use division by prime numbers, starting with the smallest prime.

5. Worked Examples

Example 1: Prime number

Write 13 as a product of prime numbers.

First, ask: Is 13 prime?

Yes. Its only factors are 1 and 13.

So 13 is already prime. Its prime factorization is simply:

$$13=13$$

This matches the theorem because numbers greater than 1 can be prime.

Example 2: A small composite number

Find the prime factorization of 24.

Start breaking 24 into factors:

$$24=2\times 12$$

Now factor 12:

$$12=2\times 6$$

Now factor 6:

$$6=2\times 3$$

Now all the factors are prime. So:

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

You can also write this using exponents:

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

This means there are three 2s multiplied together.

Example 3: Showing uniqueness

Let’s look at the number 30.

One way to factor it is:

$$30=5\times 6=5\times 2\times 3$$

Another way is:

$$30=10\times 3=2\times 5\times 3$$

These may look different at first, but both give the same prime factors:

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

The order changed, but the prime factors did not. That is the uniqueness part of the theorem.

Example 4: A larger number with exponents

Find the prime factorization of 84.

Start with a factor pair:

$$84=2\times 42$$

Factor 42:

$$42=2\times 21$$

Factor 21:

$$21=3\times 7$$

Now every factor is prime, so:

$$84=2\times 2\times 3\times 7$$

Using exponents, this is:

$$84=2^2\times 3\times 7$$

6. Why This Theorem Matters

The Fundamental Theorem of Arithmetic helps us understand how numbers are built.

It is useful when we:

• find all the factors of a number,
• find the greatest common factor (GCF),
• find the least common multiple (LCM),
• simplify fractions,
• work with exponents.

For example, if you know:

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

and

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

it becomes easier to compare the numbers and find common factors.

7. Important Ideas to Remember

• Every number greater than 1 is either prime or composite.
• Every composite number can be written as a product of primes.
• This prime factorization is unique, except for order.
• Prime numbers are the basic building blocks of whole numbers.
• Exponents help write repeated prime factors more clearly.

8. Common Mistakes

Mistake 1: Using non-prime factors in the final answer

For example, writing:

$$18=2\times 9$$

is not a prime factorization, because 9 is not prime.

The correct prime factorization is:

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

Mistake 2: Thinking different orders mean different factorizations

These are the same:

$$2\times 3\times 5$$

and

$$5\times 2\times 3$$

The order does not matter in multiplication.

Mistake 3: Calling 1 a prime number

The number 1 is not prime. A prime number must have exactly two factors. The number 1 has only one factor.

9. Quick Practice to Think About

Try these on your own:

1. Is 17 prime or composite?
2. Write 20 as a product of prime numbers.
3. Write 45 as a product of prime numbers.
4. Explain why \(2\times 2\times 5\) and \(5\times 2\times 2\) are the same prime factorization.

Answers:

1. 17 is prime.
2. $$20=2\times 2\times 5=2^2\times 5$$
3. $$45=3\times 3\times 5=3^2\times 5$$
4. They use the same prime factors; only the order is different.

10. Brief Summary

The Fundamental Theorem of Arithmetic says that every whole number greater than 1 is either prime or can be written as a product of prime numbers.

It also says this factorization is unique, except for the order of the factors.

When you break a number into prime factors, you are finding its most basic multiplication form.

This idea is a foundation for many other number skills in math.

Prime factorization means writing a number as a product of prime numbers. A prime number has exactly two factors: 1 and itself.

This lesson will show you how to find prime factorizations using factor trees and repeated division. You will also learn how to write repeated prime factors using exponents.

Prime factorization is useful because it helps us understand how numbers are built. It can also help later with topics like finding common factors, least common multiples, and simplifying problems with large numbers.

First, remember these two types of numbers:

• Prime numbers: 2, 3, 5, 7, 11, 13, ...
• Composite numbers: numbers that have more than two factors, like 4, 6, 8, 9, 10, and 12

The number 1 is not prime and not composite.

Important idea: Every composite number can be written as a product of prime numbers in only one way, aside from the order of the factors. This is why prime factorization is so powerful.

Method 1: Using a factor tree

A factor tree starts with the number you want to factor. Then you break it into two factors. If one of those factors is composite, keep breaking it apart until all the branches end in prime numbers.

1. Start with the number.
2. Split it into any two factors.
3. Keep factoring composite numbers.
4. Stop when every branch ends in a prime number.
5. Multiply the prime factors together to check your work.

Method 2: Using repeated division

In repeated division, you divide the number by prime numbers one step at a time.

1. Start with the smallest prime number, 2.
2. If the number is divisible by 2, divide.
3. Keep dividing by prime numbers: 2, 3, 5, 7, and so on.
4. Stop when the quotient is 1.
5. The prime numbers you divided by are the prime factorization.

Helpful divisibility rules

• A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.
• A number is divisible by 10 if it ends in 0.

Writing prime factorization with exponents

If the same prime factor appears more than once, you can write it with an exponent.

For example,

\(2 \times 2 \times 2 = 2^3\)

and

\(3 \times 3 = 3^2\)

So if a number has prime factorization

\(2 \times 2 \times 2 \times 3 \times 3\)

you can write it as

$$2^3 \times 3^2$$

This form is shorter and easier to read.

Worked Example 1: Prime factorization of 18 using a factor tree

Start with 18. One way to split it is

$$18 = 2 \times 9$$

The number 2 is prime, but 9 is composite. Factor 9:

$$9 = 3 \times 3$$

Now all the factors are prime:

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

Using exponents, this is

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

Check: \(2 \times 3 \times 3 = 18\)

Worked Example 2: Prime factorization of 48 using repeated division

Start dividing by the smallest prime number possible.

$$48 \div 2 = 24$$

$$24 \div 2 = 12$$

$$12 \div 2 = 6$$

$$6 \div 2 = 3$$

$$3 \div 3 = 1$$

The prime factors are

$$2 \times 2 \times 2 \times 2 \times 3$$

Using exponents,

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

Worked Example 3: Different factor tree, same answer

Find the prime factorization of 60.

You might start with

$$60 = 6 \times 10$$

Then factor each part:

$$6 = 2 \times 3$$

$$10 = 2 \times 5$$

So

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

Reorder the primes:

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

Using exponents,

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

You could have started with different factors, like \(4 \times 15\), and you would still get the same prime factorization.

Worked Example 4: Using prime factorization to understand a number

Find the prime factorization of 72 and write it with exponents.

Use a factor tree:

$$72 = 8 \times 9$$

Now factor each part:

$$8 = 2 \times 2 \times 2$$

$$9 = 3 \times 3$$

So

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

Using exponents,

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

This tells us that 72 is made from three 2s and two 3s multiplied together.

How prime factorization is applied

Prime factorization helps you see the building blocks of a number.

• It shows exactly which prime numbers make the number.
• It helps organize repeated multiplication.
• It makes large products easier to write with exponents.
• It helps compare numbers by their prime factors.

For example, compare 36 and 54.

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

$$54 = 2 \times 3^3$$

Both numbers have 2 and 3 as prime factors, but the exponents are different. This helps us understand how the numbers are alike and how they are different.

Tips for success

• Always stop only when all factors are prime.
• If you see the same prime factor many times, use an exponent.
• Check your answer by multiplying the prime factors back together.
• Order the prime factors from least to greatest to keep your work neat.

Common mistakes to avoid

• Stopping too early, such as writing \(12 = 3 \times 4\). This is not a prime factorization because 4 is not prime.
• Using 1 as a prime factor. The number 1 is not prime.
• Forgetting repeated factors. For example, \(24\) is not \(2 \times 3 \times 4\); its prime factorization is \(2^3 \times 3\).
• Writing the wrong exponent. For example, \(2 \times 2 \times 2\) is \(2^3\), not \(2^2\).

Quick practice questions

1. Write the prime factorization of 20.
2. Write the prime factorization of 45 using exponents.
3. Write the prime factorization of 84 using any method.

Answers

1. $$20 = 2 \times 2 \times 5 = 2^2 \times 5$$
2. $$45 = 3 \times 3 \times 5 = 3^2 \times 5$$
3. $$84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$$

Summary

Prime factorization means writing a number as a product of prime numbers. You can find it by using a factor tree or by repeated division. When a prime factor repeats, use an exponent to write it more simply, such as \(2 \times 2 \times 2 = 2^3\).

As you practice, remember to keep factoring until every factor is prime. Then check by multiplying your prime factors to make sure you get the original number.

Greatest Common Factor (GCF) and the Euclidean Algorithm

When we compare two whole numbers, we sometimes want to know the largest number that divides both of them exactly. That number is called the Greatest Common Factor, or GCF.

For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The biggest factor they share is 6, so the GCF of 12 and 18 is 6.

We write this as \(\text{GCF}(12,18)=6\).

Knowing the GCF helps us in many math problems. We use it to simplify fractions, to factor expressions, and to solve word problems involving equal groups.

What is a factor?

A factor of a number is a whole number that divides it with no remainder. For example, 5 is a factor of 20 because \(20 \div 5 = 4\).

A common factor is a factor that two or more numbers share. The greatest common factor is the largest one.

Three ways to find the GCF

There are several ways to find the GCF. In this lesson, we will use these methods:

• Listing factors
• Prime factorization
• The Euclidean Algorithm

Each method can help, depending on the numbers you are working with.

Method 1: Listing factors

In this method, list all the factors of each number. Then find the largest factor that appears in both lists.

This method works well for smaller numbers.

Worked Example 1: Find \(\text{GCF}(24, 36)\) by listing factors

First, list the factors of 24:

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

Now list the factors of 36:

\(36: 1, 2, 3, 4, 6, 9, 12, 18, 36\)

The common factors are 1, 2, 3, 4, 6, and 12.

The greatest common factor is 12.

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

Method 2: Prime factorization

A prime number has exactly two factors: 1 and itself. Examples are 2, 3, 5, 7, and 11.

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

For example,

To find the GCF using prime factorization:

1. Write each number as a product of prime factors.
2. Circle or identify the prime factors the numbers have in common.
3. Multiply the common prime factors.

If a prime appears more than once in both numbers, use only the number of times it appears in both.

Worked Example 2: Find \(\text{GCF}(48, 60)\) using prime factorization

First, factor each number into primes:

Now find the prime factors they share:

• Both have two 2s
• Both have one 3

So the common prime factors are:

Multiply:

So, \(\text{GCF}(48,60)=12\).

Why the GCF is useful

One common use of the GCF is simplifying fractions. To simplify a fraction, divide the numerator and denominator by their GCF.

For example, simplify \(\frac{18}{24}\).

The GCF of 18 and 24 is 6. Divide both by 6:

Method 3: The Euclidean Algorithm

The Euclidean Algorithm is a fast way to find the GCF of larger numbers. It uses division and remainders.

The main idea is this:

If one number does not divide the other evenly, divide and look at the remainder. Then replace the larger number with the smaller number, and replace the smaller number with the remainder. Keep going until the remainder is 0. The last nonzero remainder is the GCF.

This may sound tricky at first, but the steps are very organized.

Steps for the Euclidean Algorithm

1. Divide the larger number by the smaller number.
2. Write the quotient and remainder.
3. Replace the pair of numbers with the smaller number and the remainder.
4. Repeat until the remainder is 0.
5. The last nonzero remainder is the GCF.

Worked Example 3: Find \(\text{GCF}(84, 30)\) using the Euclidean Algorithm

Start by dividing 84 by 30:

This means:

Now use 30 and 24.

This means:

Now use 24 and 6.

Since the remainder is now 0, stop. The last nonzero remainder was 6.

So, \(\text{GCF}(84,30)=6\).

Worked Example 4: Find \(\text{GCF}(252, 198)\) using the Euclidean Algorithm

These numbers are larger, so the Euclidean Algorithm is very helpful.

First divide 252 by 198:

So,

Now divide 198 by 54:

So,

Now divide 54 by 36:

So,

Now divide 36 by 18:

Stop here. The last nonzero remainder is 18.

So, \(\text{GCF}(252,198)=18\).

How do you know which method to use?

• Use listing factors for small numbers.
• Use prime factorization when the numbers factor nicely into primes.
• Use the Euclidean Algorithm for larger numbers or when listing factors would take too long.

Important idea: The GCF always divides both numbers

If \(d\) is the GCF of two numbers, then both numbers can be written as multiples of \(d\).

For example, if \(\text{GCF}(48,60)=12\), then:

This is why the GCF is useful for making equal groups and simplifying.

Common mistakes to avoid

• Mixing up GCF and LCM: The GCF is the greatest factor shared by the numbers, not the smallest common multiple.
• Missing factors: When listing factors, be careful not to leave any out.
• Using too many prime factors: In prime factorization, only use the primes both numbers share.
• Stopping too early in the Euclidean Algorithm: Keep going until the remainder is 0.

Quick check

Try these on your own:

1. Find \(\text{GCF}(16, 24)\).
2. Find \(\text{GCF}(45, 75)\).
3. Find \(\text{GCF}(91, 65)\) using the Euclidean Algorithm.

Answers:

• \(\text{GCF}(16,24)=8\)
• \(\text{GCF}(45,75)=15\)
• \(91 \div 65 = 1\) remainder 26, then \(65 \div 26 = 2\) remainder 13, then \(26 \div 13 = 2\) remainder 0, so \(\text{GCF}(91,65)=13\)

Summary

The Greatest Common Factor is the largest whole number that divides two or more numbers exactly. You can find it by listing factors, using prime factorization, or using the Euclidean Algorithm.

The Euclidean Algorithm is especially useful for larger numbers because it uses repeated division with remainders. Once you understand these methods, you can choose the one that fits the problem best.

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

This is useful in real life when events repeat in cycles. For example, if one light flashes every 4 seconds and another flashes every 6 seconds, the LCM tells us when they will flash together again.

LCM is also important in fractions. When we add or compare fractions, we often need a common denominator. The least common denominator is based on the LCM of the denominators.

What does “multiple” mean?

A multiple of a number is the result of multiplying that number by whole numbers.

• 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 both lists.

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

The least common multiple is the smallest of these common multiples. So the LCM of 3 and 5 is \(15\).

How to find the LCM

There are two main methods you can use in 7th Grade:

1. List the multiples
2. Use prime factorization

Method 1: List the multiples

Write the multiples of each number until you find the first one they have in common.

This method works well when the numbers are small.

Worked Example 1: Find the LCM of 4 and 6

List the multiples:

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

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

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

Method 2: Use prime factorization

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

Then take each prime factor that appears, using the greatest number of times it appears in any one factorization.

This method is very helpful for larger numbers or more than two numbers.

Worked Example 2: Find the LCM of 12 and 18

First, write each number in prime factor form:

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

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

Now choose the greatest power of each prime:

• For 2: greatest power is \(2^2\)
• For 3: greatest power is \(3^2\)

Multiply them:

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

So the LCM is \(36\).

Why do we use the greatest powers?

The LCM must be a multiple of both numbers. That means it must include all the prime factors needed to build each number.

If one number needs \(2^2\), then the LCM must include \(2^2\). If one number needs \(3^2\), then the LCM must include \(3^2\).

That is why we take the greatest number of each prime factor.

Worked Example 3: Find the LCM of 8, 12, and 15

Prime factorize each number:

$$8=2^3$$

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

$$15=3\times 5$$

Now take the greatest power of each prime that appears:

• For 2: \(2^3\)
• For 3: \(3\)
• For 5: \(5\)

Multiply:

$$\text{LCM}(8,12,15)=2^3\times 3\times 5=8\times 3\times 5=120$$

So the LCM is \(120\).

Using LCM in a real-life situation

Suppose one bus arrives every 10 minutes and another bus arrives every 15 minutes. If they both arrive now, when will they arrive together again?

We need the LCM of 10 and 15.

Prime factorize:

$$10=2\times 5$$

$$15=3\times 5$$

Take the greatest powers:

$$\text{LCM}(10,15)=2\times 3\times 5=30$$

So the buses will arrive together again in \(30\) minutes.

LCM and fractions

When adding fractions with different denominators, we often need the least common denominator. This is the LCM of the denominators.

For example, to add \(\frac{1}{4}\) and \(\frac{1}{6}\), we find the LCM of 4 and 6.

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

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

Then rewrite the fractions:

$$\frac{1}{4}=\frac{3}{12} \qquad \frac{1}{6}=\frac{2}{12}$$

Now add:

$$\frac{3}{12}+\frac{2}{12}=\frac{5}{12}$$

Tips for finding the LCM

• The LCM must be a multiple of each number.
• The LCM is always at least as large as the greatest number in the set.
• If one number is already a multiple of the other, the larger number is the LCM.

For example, with 5 and 20, since 20 is a multiple of 5, $$\text{LCM}(5,20)=20$$

Common mistakes to avoid

• Mixing up LCM and GCF: The LCM is the smallest common multiple, while the GCF is the greatest common factor.
• Stopping too early when listing multiples: Keep going until you find the first common one.
• Using all prime factors too many times: In prime factorization, use the greatest power of each prime, not every copy from every number added together.

Quick check

Try these on your own:

1. Find the LCM of 3 and 8.
2. Find the LCM of 9 and 12.
3. Find the LCM of 6, 8, and 10.

Answers:

1. \(24\)
2. \(36\)
3. \(120\)

Summary

The least common multiple is the smallest number that two or more numbers divide into evenly.

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

LCM is helpful for solving cycle problems and finding common denominators in fractions.

When using prime factorization, remember to choose the greatest power of each prime factor.

Coprime Integers are two whole numbers that have only one common factor: 1.

This means the numbers do not share any factor bigger than 1. Even if each number has several factors, they are coprime as long as 1 is the only factor they both have.

For example, 8 and 15 are coprime.

• The factors of 8 are: 1, 2, 4, 8
• The factors of 15 are: 1, 3, 5, 15

The only factor they share is 1, so 8 and 15 are coprime.

It is important to know that coprime does not mean both numbers are prime.

For example, 8 is not prime and 15 is not prime, but they are still coprime because they share no common factor other than 1.

Main Idea: To decide if two numbers are coprime, check whether they have any common factors besides 1.

You can do this in a few ways:

• List the factors of both numbers and compare them.
• Think about divisibility: do both numbers divide by 2, 3, 5, or another number?
• Find the greatest common factor (GCF). If the GCF is 1, the numbers are coprime.

Why coprime integers matter:

• They help us understand number relationships.
• They are important when simplifying fractions.
• A fraction is in simplest form when its numerator and denominator are coprime.

For example, consider the fraction \(\frac{6}{35}\).

• Factors of 6: 1, 2, 3, 6
• Factors of 35: 1, 5, 7, 35

Since the only common factor is 1, 6 and 35 are coprime. So \(\frac{6}{35}\) is already in simplest form.

Now let’s look at some worked examples.

Example 1: Are 9 and 28 coprime?

Step 1: List the factors.

• Factors of 9: 1, 3, 9
• Factors of 28: 1, 2, 4, 7, 14, 28

Step 2: Find the common factors.

The only common factor is 1.

Answer: Yes, 9 and 28 are coprime.

Example 2: Are 14 and 21 coprime?

Step 1: List the factors.

• Factors of 14: 1, 2, 7, 14
• Factors of 21: 1, 3, 7, 21

Step 2: Find the common factors.

They share 1 and 7.

Because they share a factor greater than 1, they are not coprime.

Answer: No, 14 and 21 are not coprime.

Example 3: Use coprime integers to simplify a fraction.

Simplify \(\frac{12}{18}\).

Step 1: Find a common factor of 12 and 18.

Both numbers are divisible by 6.

Step 2: Divide numerator and denominator by 6.

Step 3: Check if 2 and 3 are coprime.

• Factors of 2: 1, 2
• Factors of 3: 1, 3

The only common factor is 1, so 2 and 3 are coprime.

Answer: \(\frac{12}{18}\) simplifies to \(\frac{2}{3}\), and this is simplest form.

Example 4: Decide whether a fraction is already simplified.

Look at \(\frac{7}{20}\).

Check whether 7 and 20 are coprime.

• Factors of 7: 1, 7
• Factors of 20: 1, 2, 4, 5, 10, 20

The only common factor is 1.

Answer: 7 and 20 are coprime, so \(\frac{7}{20}\) is already in simplest form.

Helpful Tips:

• If two numbers are both even, they are not coprime because they both have 2 as a factor.
• If two numbers both divide by 5, they are not coprime.
• Consecutive numbers, like 11 and 12, are always coprime.

Here is why consecutive numbers are coprime: if one number is 1 more than the other, they cannot both be divided evenly by the same whole number greater than 1.

Quick Check:

1. Are 5 and 12 coprime?
2. Are 18 and 27 coprime?
3. Is \(\frac{9}{16}\) in simplest form?

Answers:

1. Yes. Factors of 5 are 1, 5; factors of 12 are 1, 2, 3, 4, 6, 12. Only 1 is common.
2. No. Both share 9 and 3.
3. Yes. 9 and 16 share only 1, so the fraction is in simplest form.

Summary: Two integers are coprime if their only common factor is 1. To check, compare factors or find the greatest common factor. Coprime integers are especially useful when deciding whether a fraction is fully simplified.

Perfect Squares and Square Roots

In this lesson, you will learn what perfect squares are and how to find square roots. These ideas are important in number theory and exponents, and they also connect to geometry because squares have equal side lengths.

A square number, also called a perfect square, is a number made by multiplying a whole number by itself.

For example:

• \(1 \times 1 = 1\), so 1 is a perfect square.

• \(2 \times 2 = 4\), so 4 is a perfect square.

• \(3 \times 3 = 9\), so 9 is a perfect square.

• \(4 \times 4 = 16\), so 16 is a perfect square.

We can also write perfect squares using exponents. For example, \(5 \times 5\) can be written as \(5^2\). The exponent 2 means “multiply the number by itself once.”

Here are some common perfect squares you should know:

• \(1^2 = 1\)

• \(2^2 = 4\)

• \(3^2 = 9\)

• \(4^2 = 16\)

• \(5^2 = 25\)

• \(6^2 = 36\)

• \(7^2 = 49\)

• \(8^2 = 64\)

• \(9^2 = 81\)

• \(10^2 = 100\)

• \(11^2 = 121\)

• \(12^2 = 144\)

Notice that not every number is a perfect square. For example, 10 is not a perfect square because there is no whole number that multiplies by itself to make 10.

A square root does the opposite of squaring a number. If squaring means multiplying a number by itself, then finding a square root means asking, “What number times itself equals this number?”

The square root symbol is \(\sqrt{\phantom{x}}\).

For example:

• \(\sqrt{9} = 3\) because \(3 \times 3 = 9\)

• \(\sqrt{25} = 5\) because \(5 \times 5 = 25\)

• \(\sqrt{64} = 8\) because \(8 \times 8 = 64\)

This is why we say square roots are the inverse of squaring. They undo each other.

Here is that idea written both ways:

Perfect squares and square roots also connect to the area of a square. If a square has side length 4, then its area is:

So a square with area 16 has side length \(\sqrt{16} = 4\).

This is a helpful way to think about square roots: the square root of an area tells you the side length of the square.

Main Ideas to Remember

• A perfect square is the product of a whole number times itself.

• Squaring a number means raising it to the power of 2.

• A square root asks for the number that was multiplied by itself.

• Only some numbers are perfect squares.

• If \(a^2 = b\), then \(\sqrt{b} = a\).

How to Tell if a Number is a Perfect Square

You can use one of these strategies:

1. Think of square facts you know. For example, if you know \(7^2 = 49\), then 49 is a perfect square.

2. Check nearby squares. For example, to test 50, notice that \(7^2 = 49\) and \(8^2 = 64\). Since 50 is between them, it is not a perfect square.

3. Use arrays or square shapes. If objects can be arranged in a square with the same number of rows and columns, the total is a perfect square.

Worked Example 1: Find a Perfect Square

Is 81 a perfect square?

Ask: what whole number multiplied by itself equals 81?

So, 81 is a perfect square, and:

Worked Example 2: Find a Square Root

Find \(\sqrt{49}\).

We need the number that times itself equals 49.

So:

Worked Example 3: Decide if a Number is a Perfect Square

Is 30 a perfect square?

Let’s look at nearby perfect squares:

The number 30 is between 25 and 36, but it is not equal to either one.

So, 30 is not a perfect square.

Worked Example 4: Use Area to Find a Side Length

A square has area 121 square units. What is the side length?

For a square, area equals side length times side length. So we need:

Since:

the side length is:

So the square’s side length is 11 units.

Common Mistakes to Avoid

• Do not confuse \(n^2\) with \(2n\). For example, \(3^2 = 9\), but \(2 \times 3 = 6\).

• Do not assume every number has a whole-number square root. For example, \(\sqrt{20}\) is not a whole number.

• Be careful to remember square facts correctly. For example, \(8^2 = 64\), not 16.

Helpful Square Root Facts

• \(\sqrt{1} = 1\)

• \(\sqrt{4} = 2\)

• \(\sqrt{9} = 3\)

• \(\sqrt{16} = 4\)

• \(\sqrt{25} = 5\)

• \(\sqrt{36} = 6\)

• \(\sqrt{49} = 7\)

• \(\sqrt{64} = 8\)

• \(\sqrt{81} = 9\)

• \(\sqrt{100} = 10\)

Quick Check

• Is 36 a perfect square? Yes, because \(6^2 = 36\).

• What is \(\sqrt{100}\)? 10

• Is 18 a perfect square? No, because it is between \(4^2 = 16\) and \(5^2 = 25\).

• A square has area 64 square units. What is its side length? 8 units

Summary

A perfect square is a number you get when you multiply a whole number by itself. A square root is the opposite operation: it tells you what number was squared.

Learning square facts like \(1^2\) through \(12^2\) will help you quickly recognize perfect squares and find square roots. You can also use square roots to find the side length of a square when you know its area.

Perfect Cubes and Cube Roots are about multiplying a number by itself three times, and then undoing that multiplication.

This idea connects to real life because cubes are 3-dimensional shapes. If a cube has side length 4 units, then its volume is found by multiplying the side length three times: \(4 \times 4 \times 4 = 64\). So, \(64\) is called a perfect cube.

In this lesson, you will learn what perfect cubes are, how to find them, and how to find cube roots.

1. What is a perfect cube?

A perfect cube is a number that can be written as a number multiplied by itself three times.

In math, we write this using an exponent of 3:

Some examples are:

• \(1^3 = 1 \times 1 \times 1 = 1\)
• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(3^3 = 3 \times 3 \times 3 = 27\)
• \(4^3 = 4 \times 4 \times 4 = 64\)
• \(5^3 = 5 \times 5 \times 5 = 125\)
• \(6^3 = 6 \times 6 \times 6 = 216\)

So the numbers \(1, 8, 27, 64, 125, 216\) are all perfect cubes.

2. What is a cube root?

A cube root asks: “What number was multiplied by itself three times to make this number?”

The symbol for cube root is \(\sqrt[3]{\phantom{x}}\).

For example:

because

Cube roots and cubes are opposite operations.

• If \(4^3 = 64\), then \(\sqrt[3]{64} = 4\)
• If \(2^3 = 8\), then \(\sqrt[3]{8} = 2\)

3. Perfect cubes and volume

A cube has length, width, and height that are all the same. To find the volume of a cube, use:

where \(V\) is volume and \(s\) is the side length.

Example: If a cube has side length \(3\) cm, then

If you know the volume and want the side length, use the cube root.

Example: If the volume is \(125\text{ cm}^3\), then the side length is

4. How to recognize perfect cubes

To check if a number is a perfect cube, ask whether it matches one of the cube values of whole numbers.

It helps to memorize the first few perfect cubes:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• \(6^3 = 216\)
• \(7^3 = 343\)
• \(8^3 = 512\)
• \(9^3 = 729\)
• \(10^3 = 1000\)

If a number is in this pattern, it is a perfect cube. If not, then it is not a perfect cube of a whole number.

For example:

• \(64\) is a perfect cube because \(4^3 = 64\)
• \(30\) is not a perfect cube because no whole number multiplied three times gives \(30\)

5. Worked examples

Example 1: Find the perfect cube

Find \(3^3\).

Multiply \(3\) by itself three times:

Answer: \(27\)

Example 2: Find the cube root

Find \(\sqrt[3]{216}\).

Ask: what number multiplied by itself three times equals \(216\)?

We know:

So,

Answer: \(6\)

Example 3: Decide whether a number is a perfect cube

Is \(100\) a perfect cube?

Look at the nearby perfect cubes:

• \(4^3 = 64\)
• \(5^3 = 125\)

Since \(100\) is between \(64\) and \(125\), and it is not equal to either one, it is not a perfect cube.

Answer: No, \(100\) is not a perfect cube.

Example 4: Use cube root with volume

A cube has a volume of \(343\text{ cm}^3\). What is its side length?

Use the cube root:

Since

the side length is \(7\) cm.

Answer: \(7\text{ cm}\)

6. Common mistakes to avoid

• Do not confuse squaring and cubing. \(4^2 = 16\), but \(4^3 = 64\).
• Remember that cubing means multiplying three equal factors. For example, \(2^3 = 2 \times 2 \times 2\), not \(2 \times 3\).
• Cube root means undoing a cube. If \(x^3 = 125\), then \(x = 5\).
• Check the exact value. A number close to a perfect cube is not the same as being a perfect cube.

7. Quick practice ideas

Try asking yourself these questions:

• What is \(2^3\)?
• What is \(5^3\)?
• What is \(\sqrt[3]{64}\)?
• Is \(50\) a perfect cube?
• A cube has volume \(512\text{ cm}^3\). What is its side length?

8. Summary

A perfect cube is a number made by multiplying the same whole number three times. A cube root tells you what number was multiplied by itself three times.

Perfect cubes include \(1, 8, 27, 64, 125, 216\), and more. Cube roots are useful in number patterns and in finding the side length of a cube when you know its volume.

Properties of Integer Exponents help us simplify expressions with powers. An exponent tells how many times a base is multiplied by itself. For example, in \(2^4\), the base is 2 and the exponent is 4, so \(2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16\).

When we work with exponents, there are several important rules. These rules make it much easier to simplify expressions without writing out every multiplication.

1. Product of Powers Rule

If the bases are the same and we multiply, we add the exponents:

This works because we are combining repeated multiplication of the same base.

Example:

Check by expanding:

2. Quotient of Powers Rule

If the bases are the same and we divide, we subtract the exponents:

This rule works when \(a \ne 0\).

Example:

Check by expanding:

3. Power of a Power Rule

When a power is raised to another power, multiply the exponents:

Example:

This means \(2^3\) is being multiplied by itself 4 times:

4. Power of a Product Rule

If a product is raised to a power, raise each factor to that power:

Example:

5. Power of a Quotient Rule

If a quotient is raised to a power, raise the numerator and denominator to that power:

Example:

6. Zero Exponent Rule

Any nonzero number raised to the 0 power equals 1:

Examples:

• \(7^0 = 1\)
• \((-3)^0 = 1\)
• \(100^0 = 1\)

This can be understood using the quotient rule:

But any number divided by itself is 1, so \(a^0 = 1\).

7. Negative Exponent Rule

A negative exponent does not mean the answer is negative. It means take the reciprocal:

Examples:

• \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
• \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\)

If the negative exponent is in the denominator, it can move to the numerator and become positive:

Important Notes to Remember

• These rules work best when the bases are the same.
• \(2^3 + 2^2\) cannot be combined using exponent rules because it is addition, not multiplication.
• Be careful with negative numbers and parentheses.

For example, compare these two expressions:

• \(-2^2 = -(2^2) = -4\)
• \((-2)^2 = 4\)

The parentheses change the meaning.

Worked Example 1: Multiply powers with the same base

Simplify \(4^3 \cdot 4^2\).

Use the product of powers rule:

Now evaluate:

Answer: \(1024\)

Worked Example 2: Divide powers with the same base

Simplify \(\frac{7^5}{7^3}\).

Use the quotient of powers rule:

Now evaluate:

Answer: \(49\)

Worked Example 3: Power of a power and zero exponent

Simplify \(\left(3^2\right)^3 \cdot 5^0\).

First use the power of a power rule:

Next use the zero exponent rule:

Now multiply:

Answer: \(729\)

Worked Example 4: Negative exponents

Simplify \(\frac{2^3}{2^5}\).

Use the quotient of powers rule:

A negative exponent means reciprocal:

Answer: \(\frac{1}{4}\)

Common Mistakes

• Mistake: \(a^m + a^n = a^{m+n}\)
  This is false. Exponent rules do not work with addition.
• Mistake: Thinking \(a^{-n}\) is negative.
  It really means the reciprocal: \(a^{-n} = \frac{1}{a^n}\).
• Mistake: Forgetting that \(a^0 = 1\) for any nonzero \(a\).
• Mistake: Ignoring parentheses with negative numbers.

Quick Rule List

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\), \(a \ne 0\)
• Power of a power: \(\left(a^m\right)^n = a^{mn}\)
• Power of a product: \((ab)^n = a^n b^n\)
• Power of a quotient: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), \(b \ne 0\)
• Zero exponent: \(a^0 = 1\), \(a \ne 0\)
• Negative exponent: \(a^{-n} = \frac{1}{a^n}\), \(a \ne 0\)

Summary

Exponent rules help us rewrite and simplify expressions quickly. When multiplying powers with the same base, add exponents. When dividing, subtract exponents. When raising a power to a power, multiply exponents.

Also remember that any nonzero number to the 0 power is 1, and a negative exponent means take the reciprocal. If you watch for same bases and use parentheses carefully, exponent problems become much easier.

Operations with Scientific Notation help us work with very large numbers and very small numbers in a neat, organized way.

Scientific notation writes a number as:

$$a \times 10^n$$

where (a) is at least 1 but less than 10, and (n) is an integer.

For example:

• (3.2 \times 10^4 = 32{,}000)
• (5.7 \times 10^{-3} = 0.0057)

In this lesson, you will learn how to multiply, divide, add, and subtract numbers written in scientific notation.

First, remember the basic rule for proper scientific notation:

• The first factor must be a number from 1 up to, but not including, 10.
• The second factor is a power of 10.

So these are in scientific notation:

• (4.5 \times 10^6)
• (9.01 \times 10^{-2})

But these are not in correct scientific notation:

• (12.3 \times 10^4) because 12.3 is too large
• (0.4 \times 10^7) because 0.4 is too small

1. Multiplying in Scientific Notation

To multiply numbers in scientific notation, do two things:

1. Multiply the decimal parts.
2. Add the exponents.

This works because of the exponent rule:

$$10^a \times 10^b = 10^{a+b}$$

After multiplying, check whether your answer is still in proper scientific notation. If not, adjust it.

Worked Example 1

Multiply:

$$\left(2 \times 10^3\right)\left(4 \times 10^5\right)$$

Step 1: Multiply the decimal parts.

$$2 \times 4 = 8$$

Step 2: Add the exponents.

$$10^3 \times 10^5 = 10^{3+5} = 10^8$$

Answer:

$$8 \times 10^8$$

This is already in correct scientific notation because 8 is between 1 and 10.

Worked Example 2

Multiply:

$$\left(6 \times 10^2\right)\left(3 \times 10^4\right)$$

Step 1: Multiply the decimal parts.

$$6 \times 3 = 18$$

Step 2: Add the exponents.

$$10^2 \times 10^4 = 10^6$$

So we get:

$$18 \times 10^6$$

But 18 is not between 1 and 10, so we must rewrite it.

$$18 = 1.8 \times 10^1$$

Now combine that extra power of 10:

$$18 \times 10^6 = 1.8 \times 10^1 \times 10^6 = 1.8 \times 10^7$$

Final answer:

$$1.8 \times 10^7$$

2. Dividing in Scientific Notation

To divide numbers in scientific notation, do two things:

1. Divide the decimal parts.
2. Subtract the exponents.

This works because:

$$\frac{10^a}{10^b} = 10^{a-b}$$

Then check whether the answer is in proper scientific notation.

Worked Example 3

Divide:

$$\frac{8 \times 10^7}{2 \times 10^3}$$

Step 1: Divide the decimal parts.

$$8 \div 2 = 4$$

Step 2: Subtract the exponents.

$$10^7 \div 10^3 = 10^{7-3} = 10^4$$

Answer:

$$4 \times 10^4$$

3. Adding and Subtracting in Scientific Notation

Adding and subtracting are different from multiplying and dividing.

You cannot just add or subtract the exponents right away.

Before adding or subtracting, the powers of 10 must be the same.

This is like adding like terms. For example, you can add 3 tens and 5 tens, but not 3 tens and 5 hundreds without rewriting one of them first.

Steps for adding or subtracting:

1. Rewrite the numbers so they have the same power of 10.
2. Add or subtract the decimal parts.
3. Keep the power of 10.
4. Rewrite if needed so the answer is in proper scientific notation.

Worked Example 4

Add:

$$3.2 \times 10^5 + 4.5 \times 10^5$$

The powers of 10 are already the same, so add the decimal parts:

$$3.2 + 4.5 = 7.7$$

Keep the power of 10:

$$7.7 \times 10^5$$

Answer:

$$7.7 \times 10^5$$

Now look at a subtraction example where the powers of 10 are different.

Worked Example 5

Subtract:

$$5.6 \times 10^6 - 2.1 \times 10^5$$

The powers of 10 are not the same, so rewrite one number.

Since:

$$2.1 \times 10^5 = 0.21 \times 10^6$$

Now the expression becomes:

$$5.6 \times 10^6 - 0.21 \times 10^6$$

Subtract the decimal parts:

$$5.6 - 0.21 = 5.39$$

Keep the power of 10:

$$5.39 \times 10^6$$

Final answer:

$$5.39 \times 10^6$$

4. How to Rewrite Numbers to Match Powers of 10

When you move the decimal one place to the left, the exponent increases by 1.

When you move the decimal one place to the right, the exponent decreases by 1.

For example:

• $$4.2 \times 10^3 = 0.42 \times 10^4$$
• $$7.8 \times 10^5 = 78 \times 10^4$$

These names represent the same value, just written in different ways.

5. About Significant Figures

When working in scientific notation, people often use scientific notation to show how precise a measurement is.

Significant figures are the digits that show the amount of detail in a number.

For 7th Grade, the most important idea is this: keep the digits given in the problem unless your teacher tells you to round.

For example:

• (3.20 \times 10^4) has three significant figures.
• (3.2 \times 10^4) has two significant figures.

If you are told to round, round only at the end unless your teacher says otherwise.

Common Mistakes to Avoid

• Do not add exponents when adding numbers. You only add exponents when multiplying.
• Do not subtract exponents when subtracting numbers. You subtract exponents when dividing.
• Make sure the first factor is between 1 and 10 in the final answer.
• Match powers of 10 before adding or subtracting.

Quick Check

• Multiply: (2 \times 10^3)(5 \times 10^2) = 10 \times 10^5 = 1 \times 10^6
• Divide: ((9 \times 10^8) \div (3 \times 10^2) = 3 \times 10^6)
• Add: (1.5 \times 10^4 + 2.5 \times 10^4 = 4.0 \times 10^4)
• Subtract: (7 \times 10^3 - 2 \times 10^3 = 5 \times 10^3)

Summary

• To multiply in scientific notation, multiply the decimal parts and add the exponents.
• To divide, divide the decimal parts and subtract the exponents.
• To add or subtract, first rewrite the numbers so they have the same power of 10.
• Always make sure your final answer is in proper scientific notation.

Scientific notation makes very large and very small numbers easier to read and calculate. Once you know which operation rule to use, the problems become much simpler.