The division method is a systematic way to find the Highest Common Factor (HCF) of two or more numbers. Here's how it works, explained step-by-step:
Steps to Find HCF using the Division Method
The division method, also known as the Euclidean algorithm, relies on repeated division until a remainder of zero is obtained. The last non-zero divisor is the HCF.
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Initial Division:
- Identify the largest and smallest numbers among the given numbers.
- Divide the largest number (dividend) by the smallest number (divisor).
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Iterative Division:
- If the remainder is not zero, the previous divisor now becomes the new dividend, and the remainder becomes the new divisor.
- Divide the new dividend by the new divisor.
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Repeat:
- Continue this process of dividing the previous divisor by the previous remainder until the remainder becomes zero.
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HCF Determination:
- The last non-zero divisor (the divisor that resulted in a remainder of zero) is the HCF of the original numbers.
Example: Finding the HCF of 48 and 18
Let's illustrate this with an example, finding the HCF of 48 and 18:
| Step | Dividend | Divisor | Remainder |
|---|---|---|---|
| 1 | 48 | 18 | 12 |
| 2 | 18 | 12 | 6 |
| 3 | 12 | 6 | 0 |
- In step 1, 48 is divided by 18, giving a remainder of 12.
- In step 2, 18 is divided by 12, giving a remainder of 6.
- In step 3, 12 is divided by 6, giving a remainder of 0.
Since the last non-zero divisor is 6, the HCF of 48 and 18 is 6.
Example with Three Numbers
To find the HCF of three numbers, first find the HCF of any two numbers, then find the HCF of the result with the third number. For example, to find the HCF of 12, 18 and 30:
- Find the HCF of 12 and 18, which is 6.
- Find the HCF of 6 and 30, which is 6.
Therefore, the HCF of 12, 18 and 30 is 6.
Key Insights
- The division method is efficient and guaranteed to produce the HCF of any two or more numbers.
- The method is based on the principle that the HCF of two numbers also divides their difference, making this iterative process effective.
- This approach is especially useful when dealing with large numbers as it's generally quicker than prime factorization.
