The HCF (Highest Common Factor) method of long division, also known as the Euclidean Algorithm, is a systematic way to find the largest number that divides exactly into two or more numbers. Here's how it works:
Steps for HCF Long Division Method
Based on the provided reference, here's a breakdown of the steps:
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Divide: Divide the larger number by the smaller number.
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New Dividend & Divisor: If there's a remainder, take the previous divisor as the new dividend and the remainder as the new divisor. Divide the previous divisor by the remainder.
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Repeat: Continue this process until the remainder is zero.
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HCF: The last divisor (the one that resulted in a remainder of zero) is the HCF of the original numbers.
Example: Finding the HCF of 24 and 15
Here’s how to find the HCF of 24 and 15 using the long division method:
| Division | Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|---|
| 1st Division | 24 | 15 | 1 | 9 |
| 2nd Division | 15 | 9 | 1 | 6 |
| 3rd Division | 9 | 6 | 1 | 3 |
| 4th Division | 6 | 3 | 2 | 0 |
Since the last divisor before the remainder became zero is 3, the HCF of 24 and 15 is 3. The reference confirms this: "Therefore, HCF of 24 and 15 is 3."
Practical Insights
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This method is particularly useful for finding the HCF of large numbers where listing all the factors would be cumbersome.
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The algorithm works because the HCF of two numbers also divides their difference. Each step essentially reduces the numbers while preserving their HCF.
Summary
The HCF long division method is an efficient and reliable technique to determine the highest common factor between two or more numbers through a series of divisions until a remainder of zero is achieved. The last non-zero divisor is the HCF.
