The common division method is a way to find the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of two or more numbers. Here’s how it works:
Understanding the Common Division Method
The common division method, also referred to as the repeated division method, relies on repeatedly identifying and dividing by common prime factors. The process continues until no more common prime factors can be found. The product of the common prime factors gives the HCF.
Steps Involved:
- Identify a Common Prime Factor: Start by finding a prime number that divides all the given numbers without leaving any remainder. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11).
- Divide by the Common Factor: Divide each of the given numbers by this common prime factor. Write the quotients (the results of the division) below the numbers.
- Obtain a Common Factor of the Resultant Quotients: Look at the new set of quotients obtained in the previous step. Check if they share any common prime factors. If they do, repeat the division using this common factor.
- Repeat Until No More Common Factors: Continue this process until you reach a stage where the resulting quotients do not have any common prime factors other than 1.
- Calculate the HCF: The Highest Common Factor (HCF) is the product of all the common prime factors you used in the division process.
Example:
Let's find the HCF of 24, 36, and 60 using the common division method:
| Step | Prime Factor | 24 | 36 | 60 |
|---|---|---|---|---|
| 1 | 2 | 12 | 18 | 30 |
| 2 | 2 | 6 | 9 | 15 |
| 3 | 3 | 2 | 3 | 5 |
- Step 1: All three numbers are divisible by 2. Dividing 24, 36 and 60 by 2, we get 12, 18 and 30 respectively.
- Step 2: The quotients 12, 18 and 30 are again divisible by 2. This leaves 6, 9 and 15.
- Step 3: 6, 9, and 15 are divisible by 3 leaving 2, 3 and 5.
- Step 4: Now, 2, 3 and 5 do not have any common factor other than 1. Thus, the process stops here.
- Step 5: The HCF is the product of all the common prime factors: 2 2 3 = 12. Therefore, the HCF of 24, 36, and 60 is 12.
Practical Insights:
- Start with the Smallest Prime: It's often easiest to start with the smallest prime number (2) and work your way up to larger primes if necessary.
- Systematic Approach: Being systematic in checking for common factors will help to minimize errors.
- Understanding Prime Numbers: A good understanding of prime numbers is essential for the successful application of this method.
- Multiple Numbers: The method works effectively for finding the HCF of two or more numbers.
This method provides a structured approach to calculating the HCF of given numbers by systematically reducing them using their common prime factors until no further common factor is available.
