The Highest Common Factor (HCF) of two consecutive numbers is always 1.
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. For any pair of consecutive integers—numbers that follow each other in order, such as 5 and 6, or 100 and 101—their HCF will consistently be one.
This fundamental property stems from the fact that consecutive numbers share no common factors other than 1. If a number d were a common factor of two consecutive integers, n and n+1, then d would have to divide their difference. The difference between n+1 and n is simply 1. Since the only positive integer that can divide 1 is 1 itself, it logically follows that 1 is the sole common factor, making it the highest.
Understanding the HCF of Consecutive Numbers
To clarify this concept, let's look at various examples of consecutive number pairs and determine their HCF.
- Definition of Consecutive Numbers: Integers that are directly next to each other on the number line, differing by precisely 1 (e.g., x and x+1).
- Factors: Numbers that divide a given number exactly, without leaving a remainder.
The table below illustrates this principle:
| First Number | Second Number | Factors of First Number | Factors of Second Number | Common Factors | Highest Common Factor (HCF) |
|---|---|---|---|---|---|
| 4 | 5 | 1, 2, 4 | 1, 5 | 1 | 1 |
| 9 | 10 | 1, 3, 9 | 1, 2, 5, 10 | 1 | 1 |
| 21 | 22 | 1, 3, 7, 21 | 1, 2, 11, 22 | 1 | 1 |
| 149 | 150 | 1, 149 | 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150 | 1 | 1 |
As the table demonstrates, irrespective of the magnitude of the consecutive numbers, the only factor they share is 1.
The Underlying Reason: Why Always 1?
The mathematical proof for why the HCF of two consecutive integers is always 1 is straightforward and elegant:
- Assume a Common Divisor: Let's assume there is a common divisor d that divides both n and n+1.
- Divisibility of Difference: If d divides n and d divides n+1, then d must also divide their difference: (n+1) - n.
- Resulting Difference: The difference (n+1) - n simplifies to 1.
- Unique Divisor of 1: The only positive integer that can divide 1 without a remainder is 1 itself.
- Conclusion: Therefore, the only common divisor for any two consecutive integers is 1, making 1 their Highest Common Factor.
This fundamental property is a key concept in elementary number theory and underscores the unique relationship between consecutive integers. Understanding HCF is valuable for simplifying mathematical expressions, particularly in fractions, and for solving various problems related to divisibility.
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