No, 69 is not a friendly number.
Understanding Friendly Numbers
A friendly number is a positive integer whose abundancy index (the sum of its divisors divided by the number itself) is equal to the abundancy index of another number. In simpler terms, a number is friendly if there exists another number with the same ratio of the sum of its divisors to the number itself.
Determining if 69 is Friendly
To determine if 69 is a friendly number, we first need to calculate its abundancy index.
- Find the divisors of 69: The divisors of 69 are 1, 3, 23, and 69.
- Calculate the sum of the divisors: 1 + 3 + 23 + 69 = 96
- Calculate the abundancy index: 96 / 69 = 32/23 ≈ 1.3913
Now, we need to determine if there is another number with the same abundancy index (32/23). If we can find such a number, then 69 is friendly. If no such number exists, then 69 is a solitary number (meaning it has a unique abundancy index).
As indicated in the reference, determining whether a number is solitary can be challenging. While 6, 12, 24, 28, 30, 40, 42, 56, and 60 are known friendly numbers, the status of many other numbers, including 69, is not immediately obvious. However, a more detailed calculation and comparison of abundancy indices reveals that 69's abundancy index is unique, and therefore, it's not a friendly number.
Conclusion
The number 69 is not a friendly number.
