What Is The Greatest Common Factor Of 72 And 18?

The highest common factor (HCF) of 18 and 72 is 18, not 18. Let’s break down why and explore the connection between HCF and LCM (Least Common Multiple).

Understanding HCF

The HCF is the largest number that divides into both numbers without leaving a remainder. Think of it as finding the biggest common factor. To find the HCF of 18 and 72, we can list the factors of each number:

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The largest common factor is 18, which is the HCF.

Understanding LCM

The LCM is the smallest number that is a multiple of both numbers. Think of it as finding the smallest number they both share as a multiple. To find the LCM of 18 and 72, we can use the following relationship:

HCF (18, 72) * LCM (18, 72) = 18 * 72

Since we know the HCF is 18, we can solve for the LCM:

18 * LCM (18, 72) = 18 * 72

LCM (18, 72) = (18 * 72) / 18 = 72

Therefore, the LCM of 18 and 72 is 72.

In summary:

* The HCF of 18 and 72 is 18.
* The LCM of 18 and 72 is 72.
* The product of the HCF and LCM is equal to the product of the original numbers.

This relationship between HCF and LCM is a fundamental concept in number theory, often used in simplifying fractions, solving problems involving ratios, and understanding divisibility.

Let’s find the greatest common factor of 72.

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. A factor is a number that divides evenly into another number. For example, 8 is a factor of 72 because 72 divided by 8 equals 9.

To find the greatest common factor, we need to identify the largest number that divides into both numbers without leaving a remainder. Since we are only looking at the factors of 72, we don’t need to compare it to another number. In this case, the greatest common factor is 72.

It’s important to remember that the greatest common factor of any number will always be that number itself. This is because it is the largest number that divides evenly into itself.

The greatest common factor (GCF) of 72 and 81 is 9. To find the GCF, we need to determine the factors of each number. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The factors of 81 are 1, 3, 9, 27, and 81. The greatest common factor is the largest number that divides both 72 and 81 evenly, which is 9.

Finding the greatest common factor (GCF) is a fundamental concept in mathematics, particularly in number theory. It’s a valuable tool for simplifying fractions, solving problems involving divisibility, and understanding the relationships between numbers.

Here’s a deeper dive into the concept of GCF:

Prime Factorization Method: One way to find the GCF is by breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11).

For 72: 72 = 2 x 2 x 2 x 3 x 3
For 81: 81 = 3 x 3 x 3 x 3

Now, identify the common prime factors and their lowest powers present in both numbers:

* Both numbers have the prime factor 3 in common. The lowest power of 3 present in both numbers is 3² (3 x 3 = 9).

Therefore, the GCF of 72 and 81 is 3² = 9.

Listing Factors Method: This method involves listing out all the factors of each number and then identifying the largest factor that is common to both numbers. As we saw earlier, the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The factors of 81 are 1, 3, 9, 27, and 81. The greatest common factor is 9, as it is the largest number present in both lists.

Euclidean Algorithm: This is a more efficient method for finding the GCF of two numbers, especially when dealing with larger numbers. The algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCF.

Example: To find the GCF of 72 and 81:

* 81 divided by 72 leaves a remainder of 9.
* 72 divided by 9 leaves a remainder of 0.

Therefore, the GCF of 72 and 81 is 9.

Understanding the GCF can be helpful in various contexts, such as simplifying fractions. For example, if you have the fraction 72/81, you can simplify it by dividing both the numerator and denominator by their GCF (9), resulting in the simplified fraction 8/9.

The greatest common factor (GCF) of 18 and 72 is 18. To find the GCF, we need to identify the factors of each number. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

The greatest common factor is the largest number that divides into both numbers without leaving a remainder. In this case, 18 is the largest number that divides into both 18 and 72, making it the GCF.

Understanding the GCF

The GCF is a helpful concept in various mathematical contexts, particularly when working with fractions and simplifying expressions. Think of it like finding the biggest piece of cake you can cut that will divide evenly between two people. The GCF represents the largest “piece” that can be divided equally between both numbers.

Finding the GCF:

There are a few ways to find the GCF:

Listing Factors: We already demonstrated this method by listing all the factors of 18 and 72 and identifying the largest shared factor.
Prime Factorization: This involves breaking down each number into its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3, and the prime factorization of 72 is 2 x 2 x 2 x 3 x 3. The GCF is found by multiplying the common prime factors, in this case, 2 x 3 x 3 = 18.
Euclidean Algorithm: This method is more efficient for larger numbers. It involves repeatedly finding remainders until you reach a remainder of 0. The last non-zero remainder is the GCF.

Understanding the GCF helps you simplify expressions, solve problems involving fractions, and work with various mathematical concepts. It’s a valuable tool in your mathematical toolkit!

The highest common factor (HCF) of 72 and 81 is 9. Let’s dive a little deeper into what the HCF is and how we find it.

The highest common factor, also known as the greatest common divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder. To find the HCF of two numbers, like 72 and 81, you can use a couple of methods:

1. Prime Factorization:

Break down each number into its prime factors:
* 72 = 2 x 2 x 2 x 3 x 3
* 81 = 3 x 3 x 3 x 3
Identify the common prime factors and their lowest powers: Both 72 and 81 share the prime factor 3, and the lowest power of 3 that appears in both factorizations is 3 x 3 = 9.

2. Listing Factors:

List the factors of each number:
* Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
* Factors of 81: 1, 3, 9, 27, 81
Identify the largest common factor: The largest factor shared by 72 and 81 is 9.

So, the HCF of 72 and 81 is 9.

Let’s explore the factors of 18!

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

But what exactly are factors? Factors are numbers that divide evenly into another number, leaving no remainder. Think of it like this: if you have 18 cookies and want to split them evenly among your friends, you can divide them into groups of 1, 2, 3, 6, 9, or 18 cookies, and everyone gets a fair share.

Let’s break it down:

1 is a factor of 18 because 18 divided by 1 equals 18 (18 / 1 = 18).
2 is a factor of 18 because 18 divided by 2 equals 9 (18 / 2 = 9).
3 is a factor of 18 because 18 divided by 3 equals 6 (18 / 3 = 6).
6 is a factor of 18 because 18 divided by 6 equals 3 (18 / 6 = 3).
9 is a factor of 18 because 18 divided by 9 equals 2 (18 / 9 = 2).
18 is a factor of 18 because 18 divided by 18 equals 1 (18 / 18 = 1).

Understanding factors is important in many areas of mathematics, like finding the greatest common factor (GCF) of two numbers or simplifying fractions. It’s also helpful in everyday life, like when dividing up tasks or splitting costs.

Let’s find out what the greatest common factor (GCF) of 72 and 30 is.

The greatest common factor is the largest number that divides into two or more numbers without leaving a remainder. To find the GCF, we need to identify the common factors of 72 and 30.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72
Factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30

The common factors of 72 and 30 are: 1, 2, 3, and 6.

The greatest of these common factors is 6. Therefore, the greatest common factor of 72 and 30 is 6.

Let’s dive a little deeper into finding the GCF:

There are a couple of ways to find the GCF of two numbers. One method is to list out all the factors of each number, as we did above. Another, more efficient way is to use prime factorization. Here’s how that works:

1. Prime Factorization: Break down each number into its prime factors.
* 72 = 2 x 2 x 2 x 3 x 3
* 30 = 2 x 3 x 5

2. Identify Common Prime Factors: Look for the prime factors that both numbers share.
* Both 72 and 30 share the prime factors 2 and 3.

3. Multiply the Common Prime Factors: Multiply the common prime factors together.
* 2 x 3 = 6

Therefore, the greatest common factor (GCF) of 72 and 30 is 6.

Let’s break down the concept of the Greatest Common Factor (GCF) and explore how it applies to the number 64.

What is the GCF of 64?

The greatest common factor (GCF) of a number is the largest number that divides into it without leaving a remainder. Think of it as finding the biggest common factor that both the number and 1 share.

In the case of 64, the GCF is 8. This is because 8 is the largest number that divides evenly into both 64 and 1.

How many factors does sixty-four have?

Factors are numbers that divide evenly into another number. To find the factors of 64, we start by listing all the numbers that divide evenly into it:

1
2
4
8
16
32
64

So, sixty-four has seven different factors.

Let’s delve deeper into finding the GCF:

Prime Factorization: One common method to determine the GCF is through prime factorization. Prime factorization involves breaking down a number into its prime factors, which are numbers greater than 1 that are only divisible by 1 and themselves.

* For example, the prime factorization of 64 is 2 x 2 x 2 x 2 x 2 x 2.

Identifying Common Factors: To find the GCF, we identify the common prime factors between the number and 1, and then multiply those factors together. In the case of 64, the only prime factor it shares with 1 is 2.

Calculating the GCF: We look for the smallest exponent of each common prime factor. Since 2 appears six times in the prime factorization of 64, the GCF is 2 raised to the power of 6, which is 2 x 2 x 2 x 2 x 2 x 2 = 8.

Therefore, 8 is the largest number that divides into both 64 and 1 without a remainder, making it the GCF of 64.

You’re right! 72 is a common multiple of 18 and 24. Let’s figure out why:

To find the least common multiple (LCM) of 18 and 24, we need to list out their multiples.

* Multiples of 18 are: 18, 36, 54, 72, 90…
* Multiples of 24 are: 24, 48, 72, 96, 120…

Notice that 72 appears in both lists! This makes 72 a common multiple of 18 and 24. It’s also the *smallest* number that appears in both lists, which makes it the *least* common multiple.

Think of it like finding a shared birthday. Imagine you have two friends, one born in February (month 2) and another born in June (month 6). They both share a birthday in December (month 12), but December is the least common multiple because it’s the smallest month where they both have a birthday.

In the case of 18 and 24, 72 is their shared birthday – the smallest number that they both “share” as a multiple.

The greatest common factor (GCF) of 18, 45, and 72 is 9.

Let’s break down how to find the GCF. The GCF is the largest number that divides evenly into all the numbers in a set. To find the GCF, we can use a method called prime factorization.

Prime factorization is the process of breaking down a number into its prime factors. A prime factor is a number that is only divisible by 1 and itself. For example, the prime factors of 18 are 2, 3, and 3. We can write this as 18 = 2 x 3 x 3.

Here’s how to find the GCF of 18, 45, and 72 using prime factorization:

1. Prime factorize each number:
– 18 = 2 x 3 x 3
– 45 = 3 x 3 x 5
– 72 = 2 x 2 x 2 x 3 x 3

2. Identify the common prime factors:
– All three numbers share the prime factors 3 and 3.

3. Multiply the common prime factors together:
– 3 x 3 = 9

Therefore, the GCF of 18, 45, and 72 is 9. This means that 9 is the largest number that can divide evenly into all three numbers.

Let’s dive into the world of multiples! You’re wondering what the multiples of 18 are. Think of it this way: multiples are the numbers you get when you multiply a number by any whole number. So, for 18, we’re multiplying it by 1, 2, 3, 4, and so on.

Here’s a list of the first few multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270. And the list goes on! You can keep adding 18 to the previous number to find the next multiple. It’s like a fun pattern that never ends.

If you want to find out if a number is a multiple of 18, you can divide it by 18. If the result is a whole number (no decimals!), then the number is a multiple of 18. For example, 90 divided by 18 is 5, a whole number, so 90 is a multiple of 18.

The greatest common factor (GCF) of 18 and 72 is 18. Let’s break down how to find this.

One simple way to determine the GCF is to list out all the factors of both numbers. Factors are numbers that divide evenly into a given number.

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The greatest common factor is the largest number that is a factor of both numbers. In this case, the largest number common to both lists is 18.

Understanding the Concept:

The greatest common factor is essentially the biggest number that can divide both 18 and 72 without leaving a remainder. Think of it like finding the biggest piece of cake you can cut that will divide equally among 18 and 72 people.

For example, you could cut the cake into 18 slices, giving each person one slice. You could also cut it into 9 slices and give each person two slices. However, you can’t cut the cake into 24 slices and have everyone get a whole slice because 18 doesn’t divide evenly by 24. The largest number of slices you can cut while ensuring everyone gets a whole slice is 18.

This concept is useful in many areas of math, including simplifying fractions, solving equations, and understanding relationships between numbers.

Finding the Greatest Common Factor (GCF) of two numbers is like finding the biggest number that divides both of them evenly. Let’s break down how to find the GCF of 72 and 18.

We can do this using long division. The GCF is the divisor that leaves a remainder of 0 after we repeatedly divide the larger number by the smaller number.

First, we divide 72 (the larger number) by 18 (the smaller number). Since 72 divided by 18 equals 4 with no remainder, we know that 18 is a common factor of both 72 and 18. And since it’s the biggest number that divides both of them evenly, it’s the GCF.

So, the GCF of 72 and 18 is 18.

Think of it like this: Imagine you have 72 cookies and 18 candies. You want to divide them into equal groups, with the largest possible number of items in each group. You can divide both the cookies and candies into groups of 18, making 4 groups of cookies and 1 group of candies. You can’t divide them into larger groups while keeping both types of treats in the same group. That’s why 18 is the GCF.

Let’s find the greatest common factor (GCF) of 72 and 40.

First, we need to list out the factors of each number:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Now, let’s identify the common factors, those that appear in both lists: 1, 2, 4, and 8.

The greatest common factor is the largest of these common factors, which is 8.

Understanding the Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides into two or more numbers without leaving a remainder. It’s useful in various math situations, including simplifying fractions and solving problems involving quantities that need to be divided into equal groups. Think of it as the biggest “chunk” that can be taken out of both numbers.

Finding the GCF

There are a few methods for finding the GCF:

1. Listing Factors: This is the method we just used. It involves listing all the factors of each number and then identifying the common ones.
2. Prime Factorization: This involves breaking down each number into its prime factors (factors that are only divisible by 1 and themselves). The GCF is then the product of the common prime factors, each raised to the lowest power it appears in either factorization.
3. Euclidean Algorithm: This is a more efficient method, especially for larger numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder until you get a remainder of 0. The last non-zero remainder is the GCF.

Example: Finding the GCF of 33,264 and 35,640

Let’s use the prime factorization method to find the GCF of 33,264 and 35,640:

Prime factorization of 33,264: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 109
Prime factorization of 35,640: 2 x 2 x 2 x 2 x 5 x 11 x 41

The common prime factors are 2 x 2 x 2 x 2 = 16. Therefore, the GCF of 33,264 and 35,640 is 16.

Let’s find the greatest common factor (GCF) of 18 and 27!

First, we need to find the factors of each number. Factors are numbers that divide evenly into a given number.

The factors of 18 are: 1, 2, 3, 6, 9, and 18.

The factors of 27 are: 1, 3, 9, and 27.

The common factors of 18 and 27 are the numbers that appear in both lists: 1, 3, and 9.

The greatest common factor (GCF) is the largest of these common factors, which is 9.

Understanding the Greatest Common Factor (GCF)

The GCF is like finding the biggest piece of cake you can cut that will fit perfectly into two different sized cakes. It’s the largest number that divides into both numbers without leaving any remainder.

Finding the GCF can be useful in many situations, especially when working with fractions or simplifying expressions. For example, if you want to simplify the fraction 18/27, you can divide both the numerator and denominator by their GCF, which is 9. This gives you the simplified fraction 2/3.

Here’s another way to think about it: Imagine you have 18 apples and 27 oranges. You want to group them into the largest possible groups where each group has the same number of apples and oranges. You can group them into groups of 9, with each group containing 2 apples and 3 oranges. This represents the GCF of 18 and 27, as it’s the largest number that can divide both 18 and 27 evenly.

Remember, the GCF is a helpful concept in math, and understanding it can make solving problems involving numbers easier.

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