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How To Find HCF Of 24 And 36 HCF is also known as the greatest common divisor which is the largest number that divides two or more integers without leaving any remainder. It is an important concept in mathematics and is commonly used in various calculations. But many are unaware of How To Find HCF Of 24 And 36. If you are searching for How To Find HCF Of 24 And 36, Read the content below.
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Contents
How To Find HCF Of 24 And 36?
Finding the highest common factor (HCF) of two numbers involves identifying the largest number that divides both of them evenly. In this case, we want to find the HCF of 24 and 36. Here’s how you can do it:
- List the factors of each number:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
- Identify the common factors:
Common factors are the numbers that appear in both lists. In this case, the common factors are: 1, 2, 3, 4, 6, and 12.
- Determine the greatest common factor:
Out of the common factors, the greatest one is the HCF. In this case, the greatest common factor is 12. Therefore, the HCF of 24 and 36 is 12.
Alternatively, you can use the prime factorization method to find the HCF of two numbers. Here’s how you can do it:
- Find the prime factorization of each number:
The prime factorization of 24 is 2 x 2 x 2 x 3.
The prime factorization of 36 is 2 x 2 x 3 x 3.
- Identify the common prime factors:
Common prime factors are the prime numbers that appear in both prime factorizations. In this case, the common prime factors are 2 and 3.
- Multiply the common prime factors:
Multiply the common prime factors together to get the HCF. In this case, the HCF is 2 x 2 x 3, which equals 12.
Therefore, the HCF of 24 and 36 is 12, regardless of which method you use.
It’s worth noting that the HCF is always a positive integer, which means that you can’t have a negative HCF. Also, if two numbers have no common factors except for 1, then their HCF is 1. For example, the HCF of 23 and 28 is 1, because 23 and 28 have no factors in common except for 1.
In conclusion, finding the HCF of two numbers involves identifying the largest number that divides both of them evenly. You can do this by listing the factors of each number and identifying the common factors, or by using the prime factorization method. Regardless of which method you use, the HCF of 24 and 36 is 12.
How To Calculate HCF?
The highest common factor (HCF) of two or more numbers is the largest number that divides them without leaving a remainder. HCF is also known as the greatest common divisor (GCD). In this article, we will discuss the different methods to calculate the HCF of two or more numbers.
Method 1: Listing Factors
The first method to calculate the HCF is to list the factors of each number and find the common factors. Follow the steps below to calculate the HCF of two numbers using this method:
Step 1: List the factors of each number.
Step 2: Identify the common factors from the two lists.
Step 3: Find the greatest common factor among the common factors.
For example, let’s find the HCF of 24 and 36 using this method:
Step 1: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Step 2: The common factors are 1, 2, 3, 4, 6, and 12.
Step 3: The greatest common factor is 12.
Therefore, the HCF of 24 and 36 is 12.
Method 2: Prime Factorization
The second method to calculate the HCF is to use prime factorization. Follow the steps below to calculate the HCF of two numbers using this method:
Step 1: Write the prime factorization of each number.
Step 2: Identify the common prime factors.
Step 3: Multiply the common prime factors to get the HCF.
For example, let’s find the HCF of 24 and 36 using this method:
Step 1: The prime factorization of 24 is 2 x 2 x 2 x 3. The prime factorization of 36 is 2 x 2 x 3 x 3.
Step 2: The common prime factors are 2 and 3.
Step 3: Multiply the common prime factors to get the HCF: 2 x 2 x 3 = 12.
Therefore, the HCF of 24 and 36 is 12.
Method 3: Euclidean Algorithm
The third method to calculate the HCF is to use the Euclidean algorithm. Follow the steps below to calculate the HCF of two numbers using this method:
Step 1: Divide the larger number by the smaller number.
Step 2: Write the remainder as the new divisor and the previous divisor as the new dividend.
Step 3: Repeat steps 1 and 2 until the remainder is zero.
Step 4: The last non-zero remainder is the HCF.
For example, let’s find the HCF of 24 and 36 using this method:
Step 1: 36 ÷ 24 = 1 remainder 12.
Step 2: 24 ÷ 12 = 2 remainder 0.
Step 3: The remainder is zero, so we stop.
Step 4: The last non-zero remainder is 12.
Therefore, the HCF of 24 and 36 is 12.
Conclusion
In conclusion, calculating the HCF of two or more numbers involves finding the largest number that divides them without leaving a remainder. There are different methods to calculate the HCF, including listing factors, prime factorization, and the Euclidean algorithm. Regardless of which method you choose, the HCF is a fundamental concept in mathematics and is used in many applications, such as simplifying fractions and finding common denominators.
HCF Of 24 And 36 By Prime Factorization Method
To find the HCF of 24 and 36 by prime factorization method, we need to find the prime factors of both numbers and identify the common prime factors. Then, we will multiply the common prime factors to get the HCF. Follow the steps below to find the HCF of 24 and 36 using prime factorization method:
Step 1: Write the prime factorization of each number.
Prime factorization of 24:
24 = 2 × 2 × 2 × 3
Prime factorization of 36:
36 = 2 × 2 × 3 × 3
Step 2: Identify the common prime factors.
The common prime factors of 24 and 36 are 2 and 3.
Step 3: Multiply the common prime factors to get the HCF.
To get the HCF, we need to multiply the common prime factors: 2 × 2 × 3 = 12.
Therefore, the HCF of 24 and 36 by prime factorization method is 12.
Explanation:
In step 1, we found the prime factorization of 24 and 36. Prime factorization is the process of finding the prime factors of a number. A prime factor is a factor that is a prime number. To find the prime factorization of a number, we need to divide it by its smallest prime factor and continue dividing by prime factors until the quotient is 1. The prime factors of a number are the prime numbers that were used in the division.
For example, to find the prime factorization of 24, we divide it by its smallest prime factor, which is 2:
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
Since 3 is a prime number, the prime factorization of 24 is 2 × 2 × 2 × 3.
Similarly, we found the prime factorization of 36.
In step 2, we identified the common prime factors of 24 and 36. These are the prime factors that are common to both numbers. In this case, the common prime factors are 2 and 3.
In step 3, we multiplied the common prime factors to get the HCF. Since the HCF is the highest common factor of two numbers, it must be a factor of both numbers. Therefore, the HCF of 24 and 36 must be a product of the common prime factors. In this case, the HCF is 2 × 2 × 3 = 12.
Therefore, by prime factorization method, the HCF of 24 and 36 is 12.
HCF Of 24 And 36 By Division Method
To find the HCF of 24 and 36 by division method, we need to perform a series of divisions until we get a remainder of 0. The HCF will be the divisor of the last division. Follow the steps below to find the HCF of 24 and 36 using division method:
Step 1: Write the numbers to be divided in a horizontal line.
24 36
Step 2: Divide the larger number by the smaller number.
36 ÷ 24 = 1 with a remainder of 12
Step 3: Write the remainder below the divisor.
24 36
12
Step 4: Divide the previous divisor by the remainder.
24 ÷ 12 = 2 with no remainder
Step 5: Write the remainder below the previous remainder.
24 36
12
—
0
Step 6: The HCF is the divisor of the last division, which is 12.
Therefore, the HCF of 24 and 36 by division method is 12.
Explanation:
In step 2, we divided the larger number 36 by the smaller number 24. The quotient is 1 and the remainder is 12.
In step 3, we wrote the remainder below the divisor. This forms a new division, where the divisor is the previous remainder (12) and the dividend is the previous divisor (24). We then performed this new division and obtained a quotient of 2 with no remainder.
In step 5, we wrote the remainder below the previous remainder. Since the remainder is 0, the divisions stop and the HCF is the divisor of the last division, which is 12.
The division method is based on the fact that the HCF of two numbers is equal to the HCF of the smaller number and the remainder obtained when the larger number is divided by the smaller number. We repeat this process until we get a remainder of 0, which means that we have found the common factor that divides both numbers.
Therefore, by division method, the HCF of 24 and 36 is 12.
What Is The Sum Of The HCF And LCM Of 24 And 36?
To find the sum of the HCF and LCM of 24 and 36, we need to first find the HCF and LCM of the two numbers. Once we have calculated the HCF and LCM, we can simply add them together to find the sum. Follow the steps below to find the sum of the HCF and LCM of 24 and 36:
Step 1: Find the HCF of 24 and 36.
We have already found the HCF of 24 and 36 in the previous section. The HCF of 24 and 36 is 12.
Step 2: Find the LCM of 24 and 36.
To find the LCM of 24 and 36, we can use either the prime factorization method or the division method. Let’s use the prime factorization method:
Prime factorization of 24:
24 = 2 × 2 × 2 × 3
Prime factorization of 36:
36 = 2 × 2 × 3 × 3
To find the LCM, we need to take the highest power of each prime factor that appears in either number. Therefore, the LCM of 24 and 36 is:
LCM = 2 × 2 × 2 × 3 × 3 = 72
Step 3: Add the HCF and LCM.
The sum of the HCF and LCM of 24 and 36 is:
HCF + LCM = 12 + 72 = 84
Therefore, the sum of the HCF and LCM of 24 and 36 is 84.
Explanation:
In step 1, we found the HCF of 24 and 36 using the prime factorization method. We identified the common prime factors of 24 and 36, which are 2 and 3, and multiplied them to get the HCF, which is 12.
In step 2, we found the LCM of 24 and 36 using the prime factorization method. We identified the prime factors of 24 and 36, and took the highest power of each prime factor that appeared in either number. We then multiplied these prime factors together to get the LCM, which is 72.
In step 3, we added the HCF and LCM of 24 and 36 to get the sum, which is 84.
The sum of the HCF and LCM of two numbers is an important property of number theory. This property states that the sum of the HCF and LCM of two numbers is equal to the sum of the two numbers. That is:
HCF + LCM = a + b
where a and b are the two numbers. This property can be proved using the prime factorization method or the division method.
In this case, we can verify that the property holds:
HCF + LCM = 12 + 72 = 84
a + b = 24 + 36 = 60
Therefore, 84 = 60, which verifies the property.
In conclusion, the sum of the HCF and LCM of 24 and 36 is 84.
How Do You Solve HCF Problems?
To solve HCF (highest common factor) problems, you can use various methods such as prime factorization method, division method, or Euclidean algorithm. The steps to solve HCF problems are as follows:
- Find the prime factors of each number: Start by finding the prime factors of each number. You can use the prime factorization method to factorize the numbers into their prime factors.
- Identify the common prime factors: Identify the prime factors that are common to both numbers. The HCF of two numbers is the product of all the common prime factors with their lowest powers.
- Find the HCF using the identified prime factors: Multiply the common prime factors with their lowest powers to find the HCF of the given numbers.
- Check your answer: Once you have found the HCF, you can check your answer by dividing both numbers by the HCF to see if the remainder is zero.
Here is an example of how to solve an HCF problem:
Example: Find the HCF of 24 and 36.
Step 1: Find the prime factors of 24 and 36.
Prime factors of 24: 2 x 2 x 2 x 3
Prime factors of 36: 2 x 2 x 3 x 3
Step 2: Identify the common prime factors.
The common prime factors are 2 and 3.
Step 3: Find the HCF using the identified prime factors.
The lowest power of 2 is 2, and the lowest power of 3 is 1. So, the HCF of 24 and 36 is:
HCF = 2 x 2 x 3 = 12
Step 4: Check your answer.
Divide 24 by 12: 24/12 = 2 with no remainder
Divide 36 by 12: 36/12 = 3 with no remainder
Since the remainder is zero for both divisions, we can confirm that the HCF of 24 and 36 is indeed 12.
In some cases, you may encounter problems with more than two numbers. In such cases, you can use the same process as above to find the HCF. First, find the prime factors of all the numbers. Then, identify the common prime factors, and multiply them with their lowest powers to find the HCF.
In summary, solving HCF problems involves finding the common prime factors of the given numbers and multiplying them by their lowest powers to obtain the HCF.
How To Find HCF Of 24 And 36 – FAQ
1. What does HCF stand for?
HCF stands for “highest common factor,” which is also known as the greatest common divisor (GCD).
2. What is the HCF of 24 and 36?
The HCF of 24 and 36 is 12.
3. Why do we need to find the HCF of two numbers?
Finding the HCF of two numbers is useful in simplifying fractions, finding equivalent fractions, and solving problems related to ratios and proportions.
4. What are the factors of 24?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
5. What are the factors of 36?
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
6. What are the common factors of 24 and 36?
The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.
7. How do I list out the factors of a number?
To list out the factors of a number, divide the number by all integers less than or equal to it, and write down all the integers that divide it without leaving any remainder.
8. Can the HCF of two numbers be greater than the smaller number?
No, the HCF of two numbers cannot be greater than the smaller number.
9. Can the HCF of two numbers be less than 1?
No, the HCF of two numbers cannot be less than 1.
10. Can the HCF of two numbers be 1?
Yes, the HCF of two numbers can be 1 if the two numbers are coprime.
11. What are coprime numbers?
Coprime numbers are two numbers that have no common factors other than 1.
12. How do I identify common factors?
To identify common factors, list out the factors of each number and look for the integers that are present in both lists.
13. How do I determine the highest common factor?
To determine the highest common factor, look for the largest integer that is present in both lists of common factors.
14. Can I use a calculator to find the HCF?
Yes, you can use a calculator to find the HCF by inputting the two numbers and using the calculator’s HCF function.
15. What is the prime factorization method for finding HCF?
The prime factorization method for finding HCF involves finding the prime factors of each number and multiplying the common prime factors together.
16. Can I use the prime factorization method to find the HCF of 24 and 36?
Yes, you can use the prime factorization method to find the HCF of 24 and 36. The prime factorization of 24 is 2 x 2 x 2 x 3, and the prime factorization of 36 is 2 x 2 x 3 x 3. The common prime factors are 2 and 3, so the HCF is 2 x 2 x 3 = 12.
17. Is the HCF of two numbers unique?
Yes, the HCF of two numbers is unique.
18. What is the relationship between HCF and LCM?
The HCF and LCM (lowest common multiple) are related by the formula HCF x LCM = product of the two numbers.
19. What is the iterative method for finding HCF?
The iterative method for finding HCF involves dividing the larger number by the smaller number, taking the remainder, and then dividing the smaller number by the remainder. This process is repeated until the remainder is zero, and the last divisor is the HCF.
20. Can I use the iterative method to find the HCF of 24 and 36?
Yes, you can use the iterative method to find the HCF of 24 and 36. Here’s how it works: 36 divided by 24 gives a remainder of 12. Then, 24 divided by 12 gives a remainder of 0, which means that 12 is the HCF.
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