<<–2/”>a href=”https://exam.pscnotes.com/5653-2/”>h2>Highest Common Factor (HCF)
Definition
The highest common factor (HCF) of two or more numbers is the largest number that divides all of them without leaving a remainder. It is also known as the greatest common divisor (GCD).
Finding the HCF
There are several methods to find the HCF of two or more numbers:
1. Prime Factorization Method:
- Step 1: Find the prime factorization of each number.
- Step 2: Identify the common prime factors.
- Step 3: Multiply the common prime factors, each raised to the lowest power it appears in any of the factorizations.
Example: Find the HCF of 12 and 18.
- Prime factorization of 12: 2 x 2 x 3
- Prime factorization of 18: 2 x 3 x 3
- Common prime factors: 2 and 3
- HCF: 2 x 3 = 6
2. Division Method:
- Step 1: Divide the larger number by the smaller number.
- Step 2: If the remainder is zero, the smaller number is the HCF.
- Step 3: If the remainder is not zero, divide the smaller number by the remainder.
- Step 4: Repeat steps 2 and 3 until the remainder is zero. The last divisor is the HCF.
Example: Find the HCF of 24 and 36.
- 36 ÷ 24 = 1 remainder 12
- 24 ÷ 12 = 2 remainder 0
- Therefore, the HCF of 24 and 36 is 12.
3. Euclidean Algorithm:
This method is based on the fact that the HCF of two numbers is the same as the HCF of the smaller number and the remainder when the larger number is divided by the smaller number.
- Step 1: Divide the larger number by the smaller number.
- Step 2: Replace the larger number with the smaller number and the smaller number with the remainder.
- Step 3: Repeat steps 1 and 2 until the remainder is zero. The last divisor is the HCF.
Example: Find the HCF of 24 and 36.
- 36 ÷ 24 = 1 remainder 12
- 24 ÷ 12 = 2 remainder 0
- Therefore, the HCF of 24 and 36 is 12.
Properties of HCF
- The HCF of two numbers is always less than or equal to the smaller number.
- The HCF of two numbers is 1 if and only if the numbers are relatively prime (co-prime).
- The HCF of two numbers is equal to their product divided by their LCM.
Applications of HCF
- Simplifying FRACTIONS: The HCF can be used to simplify fractions by dividing both the numerator and denominator by the HCF.
- Finding the greatest common measure: The HCF can be used to find the greatest common measure of two lengths, areas, or volumes.
- Solving problems involving divisibility: The HCF can be used to solve problems involving divisibility, such as finding the largest number that divides two given numbers.
Least Common Multiple (LCM)
Definition
The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them.
Finding the LCM
There are several methods to find the LCM of two or more numbers:
1. Prime Factorization Method:
- Step 1: Find the prime factorization of each number.
- Step 2: Identify all the prime factors, including duplicates.
- Step 3: Multiply the prime factors, each raised to the highest power it appears in any of the factorizations.
Example: Find the LCM of 12 and 18.
- Prime factorization of 12: 2 x 2 x 3
- Prime factorization of 18: 2 x 3 x 3
- All prime factors: 2, 2, 3, 3
- LCM: 2 x 2 x 3 x 3 = 36
2. Division Method:
- Step 1: Write the numbers in a row.
- Step 2: Find the smallest prime number that divides at least two of the numbers.
- Step 3: Divide the numbers by the prime number and write the quotients below.
- Step 4: Repeat steps 2 and 3 until all the numbers are prime.
- Step 5: Multiply all the prime numbers and the remaining numbers to get the LCM.
Example: Find the LCM of 12, 18, and 24.
| Number | 12 | 18 | 24 |
|---|---|---|---|
| 2 | 6 | 9 | 12 |
| 2 | 3 | 9 | 6 |
| 3 | 1 | 3 | 2 |
| 3 | 1 | 1 | 2 |
| 2 | 1 | 1 | 1 |
LCM: 2 x 2 x 3 x 3 x 2 = 72
3. Euclidean Algorithm:
The Euclidean algorithm can also be used to find the LCM of two numbers.
- Step 1: Find the HCF of the two numbers using the Euclidean algorithm.
- Step 2: Multiply the two numbers and divide the product by their HCF.
Example: Find the LCM of 24 and 36.
- HCF of 24 and 36 is 12.
- LCM: (24 x 36) / 12 = 72
Properties of LCM
- The LCM of two numbers is always greater than or equal to the larger number.
- The LCM of two numbers is equal to their product divided by their HCF.
- The LCM of two numbers is the same as the product of the numbers if they are relatively prime (co-prime).
Applications of LCM
- Finding the least common time: The LCM can be used to find the least common time when two events occur at regular intervals.
- Solving problems involving fractions: The LCM can be used to solve problems involving fractions, such as adding or subtracting fractions with different denominators.
- Finding the least common measure: The LCM can be used to find the least common measure of two lengths, areas, or volumes.
Relationship between Hcf And Lcm
The HCF and LCM of two numbers are closely related. The following relationship holds true:
HCF(a, b) x LCM(a, b) = a x b
This relationship can be used to find either the HCF or the LCM if the other is known.
Frequently Asked Questions (FAQs)
1. What is the difference between HCF and LCM?
The HCF is the largest number that divides two or more numbers without leaving a remainder, while the LCM is the smallest number that is a multiple of two or more numbers.
2. How do I find the HCF and LCM of more than two numbers?
You can find the HCF and LCM of more than two numbers by applying the same methods described above, but you need to consider all the numbers simultaneously.
3. What is the HCF of two prime numbers?
The HCF of two prime numbers is always 1, as they have no common factors other than 1.
4. What is the LCM of two prime numbers?
The LCM of two prime numbers is equal to their product.
5. Can the HCF and LCM of two numbers be the same?
Yes, the HCF and LCM of two numbers can be the same if the two numbers are equal.
6. What is the HCF and LCM of 0 and any other number?
The HCF of 0 and any other number is the other number, while the LCM of 0 and any other number is 0.
7. How can I use HCF and LCM in real life?
HCF and LCM have various applications in real life, such as:
- Dividing objects into equal groups: The HCF can be used to find the largest number of equal groups that can be formed from a given number of objects.
- Scheduling events: The LCM can be used to find the least common time when two events occur at regular intervals.
- Measuring lengths and volumes: The HCF and LCM can be used to find the greatest common measure and least common measure of two lengths, areas, or volumes.
8. What are some examples of HCF and LCM in real life?
- Sharing cookies: If you have 12 cookies and want to share them equally among 4 friends, you can use the HCF to find the largest number of cookies each friend can get (HCF of 12 and 4 is 4).
- Meeting friends: If you meet a friend every 3 days and another friend every 5 days, the LCM (15) will tell you how many days it will take for you to meet both friends on the same day.
9. Are there any online calculators for HCF and LCM?
Yes, there are many online calculators available that can calculate the HCF and LCM of two or more numbers. You can find these calculators by searching online for “HCF calculator” or “LCM calculator.”
10. What are some other related concepts to HCF and LCM?
Some other related concepts include:
- Relatively prime numbers: Two numbers are relatively prime (co-prime) if their HCF is 1.
- Prime factorization: Prime factorization is the process of finding the prime factors of a number.
- Euclidean algorithm: The Euclidean algorithm is a method for finding the HCF of two numbers.
Understanding HCF and LCM is essential for solving various mathematical problems and understanding real-life situations. By mastering these concepts, you can gain a deeper understanding of number theory and its applications.