If you're unfamiliar with the concept of factors, be sure to read our previous explanation before continuing on. In summary, factors are numbers that can divide evenly into a given number without leaving a remainder. In this topic, we will learn how to identify the highest common factor (HCF) between two or more numbers, also known as the greatest common factor (GCF) or greatest common divisor (GCD).

The HCF is a useful tool for simplifying fractions, equations, and problem-solving. It represents the largest number that can be divided evenly by a set of given numbers. So, let's dive into understanding the meaning of the HCF further.

The HCF can be thought of as the greatest number that can be divided by a set of given numbers. It is the largest number that can evenly divide both numbers without leaving a remainder. This can be represented as:

HCF(x,y) = a, where a is the highest common factor between x and y.

In simpler terms, the HCF is the biggest number that both x and y can be divided by without a remainder. Let's use an example to better understand this concept.

For the given numbers 14 and 21, the factors of 14 are 1, 2, 7, and 14, while the factors of 21 are 1, 3, 7, and 21. By comparing these lists, we can determine that the common factors are 1 and 7. Since 7 is the largest common factor, we can confirm that the HCF of 14 and 21 is 7, as it can divide evenly into both numbers without a remainder.

Prime numbers, by definition, only have two factors - 1 and themselves. Therefore, finding the HCF of two prime numbers would result in their only common factor being 1. Let's see this in action with the prime numbers 23 and 31.

Since 23 and 31 can only be divided by 1 and themselves, their HCF is 1. This means that the HCF of any two or more prime numbers will always be 1. Go ahead and try this with the prime numbers 3, 17, and 29 - you'll get the same result!

It's important to familiarize ourselves with the properties of the HCF, which include:

- The HCF divides each given number without a remainder.
- The HCF is a factor of each given number.
- The HCF is always less than or equal to each given number.
- The HCF of two or more prime numbers is always 1.

Next, we will explore the different methods for finding the HCF of two or more numbers.

There are three main methods for finding the HCF - listing factors method, prime factorization, and the division method. Let's take a closer look at each method and work through examples.

The listing factors method is the simplest approach to find the HCF. It involves three steps:

- List the factors of each given number.
- Find the common factors between the numbers.
- Determine the highest common factor among these common factors. This number is the HCF.

For example, let's find the HCF of 27 and 36:

**Solution:**

**Step 1:** List the factors of 27 and 36.

Factors of 27: 1, 3, 9, and 27.

Factors of 36: 1, 2, 3, 4, 6, 12, 18, and 36.

**Step 2:** The common factors of 27 and 36 are 1, 3, and 9.

**Step 3:** The highest common factor among these common factors is 9. Therefore, the HCF of 27 and 36 is 9.

Finding the highest common factor (HCF) is an important task when working with numbers. It is the largest number that can be divided evenly into a set of numbers. In this article, we will explore three different methods for finding the HCF and provide examples to help you grasp them easily.

Before we delve into the different methods for finding the HCF, let's first understand what factors are. Factors are numbers that can divide a given number without leaving any remainder. For instance, the factors of 27 are 1, 3, 9, and 27, and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

A simple method for finding the HCF of two numbers is to list their factors and then determine the common factors between them. Let's take the numbers 27 and 36 as an example.

Common factors of 27 and 36: 1, 3, 9.

Among these, the largest common factor is 9, making the HCF of 27 and 36 equal to 9.

Now, let's try finding the HCF of 7, 35, and 42.

Common factors of 7, 35, and 42: 1 and 7.

Thus, the HCF of 7, 35, and 42 is 7.

The same method can be used to find the HCF of more than two numbers. For instance, the HCF of 12, 15, 21, and 24 would be 3, as it is the largest common factor among these numbers.

Another way to find the HCF is through prime factorisation. This involves breaking down each number into its prime factors (numbers that can only be divided by themselves and 1). Let's take the example of 24 and 32.

Prime factorisation of 24: 24 = 2 x 2 x 2 x 3 = 2^{3} x 3

Prime factorisation of 32: 32 = 2 x 2 x 2 x 2 x 2 = 2^{5}

Since both numbers have three instances of the number 2, the HCF of 24 and 32 is 2 x 2 x 2 = 8.

In a similar manner, we can find the HCF of 8, 12, and 20 as 2 x 2 = 4. This method can be more efficient and less complicated than listing out all the factors, especially for larger numbers.

The division method involves using long division to find the HCF. Let's see how this works for 24 and 32.

Step 1: Divide the larger number by the smaller number. 32 ÷ 24 = 1 with a remainder of 8.

Step 2: Use the remainder as the new dividend and the divisor as the new divisor, and divide again. 24 ÷ 8 = 3 with no remainder, making the HCF equal to 8.

A similar method can be used to find the HCF of multiple numbers. For example, to find the HCF of 9, 27, 36, and 45, we would follow these steps:

Step 1: Divide the larger number by the smaller number. 36 ÷ 9 = 4 with no remainder.

Step 2: Use the remainder as the new dividend and the divisor as the new divisor, and divide again. 45 ÷ 9 = 5 with no remainder.

Since we ended up with a remainder of 0 after dividing all the numbers multiple times, the HCF is 9.

For multiple numbers, we need to group them into equal pairs and then perform long division as explained above. For example, if we want to find the HCF of 7, 21, 35, and 49, we would group them into (7, 21) and (35, 49) and then find the HCF of each pair. In this case, the HCF would be 7.

This article will discuss the step-by-step process for finding the highest common factor (HCF) using the division method. This method involves breaking down a complex problem into simpler steps to make it more manageable.

Step 1: Find the HCF of the first two given numbers using long division.

Step 2: Find the HCF of the third number and the HCF obtained in Step 1.

Step 3: Repeat Step 2 to find the HCF of the fourth number and the HCF obtained from the previous steps.

Let's look at some worked examples to understand this method better.

Example 1: Find the HCF of 40 and 50.

Solution: Step 1 - Divide 50 by 40. 50 ÷ 40 = 1 with a remainder of 10.

Step 2 - Use the remainder as the new dividend and the first divisor as the new divisor, and divide again. 40 ÷ 10 = 4 with no remainder.

Therefore, the HCF of 40 and 50 is 10.

To find the HCF (also known as the greatest common factor), we use a mathematical method known as long division. This method involves dividing the given numbers by common factors until the remainder is equal to 0, then the last divisor used is the HCF. Let's look at some examples to better understand this concept.

Solution: Step 1- To find the HCF of 40 and 50, we start by dividing 50 by 40, resulting in a remainder of 10.

Step 2- As the remainder is not 0, we make the remainder (10) the divisor and the divisor (40) the dividend.

Step 3- Now, dividing 40 by 10, we get a remainder of 0. Therefore, the divisor (10) is the HCF of 40 and 50. Thus, HCF(40, 50) = 10.

Solution: Step 1- To find the HCF of 33 and 121, we apply the long division method.

Step 2- Since the remainder is not 0, we make the remainder (22) the divisor and the divisor (33) the dividend.

Step 3- Again, the remainder is not 0, so we make the remainder (11) the divisor and the divisor (22) the dividend.

Step 4- As the remainder in Step 3 is equal to 0, the divisor (11) is the HCF of 33 and 121.

Step 5- To find the HCF of 33, 121, and 154, we make the previously found HCF (11) the divisor and 154 the dividend. Applying long division, we get a remainder of 0. Therefore, the divisor (11) is the HCF of all three numbers. Thus, HCF(33, 121, 154) = 11.

Solution: Step 1- To find the HCF of 24 and 48, we first divide 48 by 24, resulting in a remainder of 0. Therefore, the divisor (24) is the HCF of these two numbers.

Step 2- To find the HCF of 24, 48, and 63, we make the previously found HCF (24) the divisor and 63 the dividend, using the long division method. This results in a remainder of 15.

Step 3- Since the remainder is not 0, we repeat the process and make the remainder (15) the divisor and the divisor (24) the dividend.

Step 4- Again, the remainder is not 0, so we make the remainder (9) the divisor and the divisor (15) the dividend.

Step 5- Same as above, the remainder (6) is not 0, so we make the remainder (6) the divisor and the divisor (9) the dividend.

Step 6- Finally, the remainder is 0. Therefore, the divisor (3) is the HCF of these four numbers. Thus, HCF(24, 48, 63, 75) = 3.

The lowest common multiple (LCM) is the smallest number that is divisible by all given numbers. It is different from the HCF, which is the largest number that divides all given numbers evenly. The following table compares the two concepts.

- The HCF is the largest number that divides all given numbers evenly.
- The LCM is the smallest number that is divisible by all given numbers.

Despite their differences, there is a clever connection between the HCF and LCM. For two numbers, x and y, their HCF and LCM (denoted by HCF(x, y) and LCM(x, y)) are related by this formula: HCF(x, y) x LCM(x, y) = x x y. This holds true for all given sets of numbers. Let's see an example to demonstrate this.

*Example: Verify that the above formula is satisfied for the numbers 9 and 12.*

Solution: First, we find the HCF of 9 and 12 using the long division method.

Factors of 9: 1, 3, 9

Factors of 12: 1, 2, 3, 4, 6, 12

Common factors of 9 and 12: 1 and 3

Therefore, HCF(9, 12) = 3.

Next, we find the LCM of 9 and 12 using the prime factorization method.

Factors of 9: 3 x 3

Factors of 12: 2 x 2 x 3

Therefore, LCM(9, 12) = 36.

Using the formula, we can see that the product of 9 and 12 is equal to 108.

*HCF(9, 12) x LCM(9, 12) = 3 x 36 = 108*

The two products are equal, thus satisfying the formula. This formula holds true for all given sets of numbers.

If you want to understand the concept of finding the HCF of a pair of numbers, let's break it down step by step with some examples using three different methods - the listing method, prime factorization method, and the division method.

**Note: To find the HCF, you can also use the Euclidean Algorithm, but for simplicity, we will only cover these three methods in this article.**

Step 1: Begin by listing out all the factors of the two numbers given. For example, let's find the HCF of 45 and 72. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Step 2: Now, compare the two lists and find the common factors. In this case, the common factors are 1, 3, and 9.

Step 3: The largest of these common factors is 9. Therefore, the HCF of 45 and 72 using the listing method is 9.

Step 1: Begin by factoring both numbers into their prime factors. For example, let's find the HCF of 45 and 72. The prime factorization of 45 is 3 x 3 x 5. The prime factorization of 72 is 2 x 2 x 2 x 3 x 3.

Step 2: Now, identify the common prime factors in the two lists. In this case, the common prime factors are 3 and 3.

Step 3: To find the HCF, multiply the common prime factors found in Step 2. In this case, the HCF of 45 and 72 using the prime factorization method is 3 x 3 = 9.

Step 1: Begin by dividing the larger number by the smaller number. For example, let's find the HCF of 45 and 72. 72 divided by 45 gives a quotient of 1 and a remainder of 27.

Step 2: Now, divide the divisor (45) by the remainder (27). This gives a quotient of 1 and a remainder of 18.

Step 3: Continue this process until the remainder is equal to 0. In this case, the next division gives a quotient of 1 and a remainder of 9. And the final division gives a quotient of 2 and a remainder of 0.

Step 4: The last divisor used is the HCF. In this case, the HCF of 45 and 72 using the division method is 9.

The Highest Common Factor (HCF) is the largest common factor shared between two whole numbers - denoted by HCF(x, y) = a. It is a crucial concept in mathematics and is used in various calculations and problem-solving.

To find the HCF, we start by listing the factors of each number. For example, for the numbers 45 and 72, the factors are:

- 45: 1, 3, 5, 9, 15, 45
- 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Next, we find the common factors shared between the two lists. In this case, they are 1, 3, and 9. Finally, we identify the largest common factor, which is 9. Therefore, the HCF of 45 and 72 is 9.

Another method to find the HCF is by writing both numbers as a product of their prime factors. For 45, the prime factorisation is 3 x 3 x 5, while 72 can be written as 2 x 2 x 2 x 3 x 3.

Then, we list the common prime factors for both numbers, which in this case is the number 3. We multiply these common factors, which gives us 9. Hence, the HCF of 45 and 72 is 9.

We can also find the HCF using the long division method. Here, we divide the larger number (72) by the smaller one (45). Since the remainder is not 0, we make the remainder (27) the divisor and the divisor (45) the dividend. We repeat this process until the remainder is 0. In this case, it takes two more steps, and the final divisor of 9 is then the HCF. Therefore, the HCF of 45 and 72 is 9.

There are a few key things to remember about the HCF:

- The HCF of two numbers divides each given number without leaving a remainder.
- The HCF of two numbers is a factor of each given number.
- The HCF of two numbers is always less than or equal to each given number.
- The HCF of two (or more) prime numbers is always 1.

The HCF and Least Common Multiple (LCM) are related in a way that the product of these two numbers is equal to the product of the original two numbers. In other words, if the HCF of two numbers is 5 and the LCM is 40, then the product of the two numbers is also 40.

This relationship can be expressed mathematically as: HCF(x, y) x LCM(x, y) = x x y.

The HCF is the largest common divisor between a pair of numbers. In other words, it is the largest possible number that completely divides both numbers, resulting in a remainder of zero. It is an essential concept in mathematics and is used in various calculations and problem-solving techniques.

To find the HCF, there are three methods you can try - the common factor method, prime factorisation, and division method. For example, the HCF of 15 and 25 is 5.

Remember, the HCF follows a few rules:

- The HCF of two numbers divides each given number without a remainder.
- The HCF of two numbers is a factor of each given number.
- The HCF of two numbers is always less than or equal to each given number.
- The HCF of two prime numbers is always 1.

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