The Highest Common Factor (HCF) of any two consecutive numbers is always one.
Understanding HCF
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving any remainder. For example, the HCF of 12 and 18 is 6, as 6 is the largest number that divides both 12 and 18 evenly.
HCF of Consecutive Numbers
Two consecutive numbers are numbers that follow each other directly in a sequence, such as 5 and 6, or 20 and 21. According to the provided reference, consecutive numbers do not share any common factors other than 1. This is because each consecutive number is one unit greater than the last, so they will never share a factor greater than 1.
Why the HCF is Always 1
- No Common Factors: Aside from 1, two consecutive numbers will never have a common factor. For example, consider the consecutive numbers 7 and 8. The factors of 7 are 1 and 7, while the factors of 8 are 1, 2, 4, and 8. The only common factor between these two numbers is 1.
- The Explanation: The explanation is that apart from 1, the two successive integers do not have any common element. This can be explained by the fact that if a number 'n' is divisible by any number 'x' then 'n+1' will not be divisible by 'x'.
- The Result: Because the only shared factor is 1, the HCF of any two consecutive numbers will always be 1.
Examples
| Consecutive Numbers | Factors | Common Factor | HCF |
|---|---|---|---|
| 3, 4 | 3: 1, 3; 4: 1, 2, 4 | 1 | 1 |
| 10, 11 | 10: 1, 2, 5, 10; 11: 1, 11 | 1 | 1 |
| 25, 26 | 25: 1, 5, 25; 26: 1, 2, 13, 26 | 1 | 1 |
Conclusion
In summary, the HCF of any two consecutive numbers is always 1. This is because these numbers share no common factors other than 1.
