Certainly! To solve the problem, we want to determine the largest possible subset of integers from 1 to 100 such that no number in this subset is three times another number within the same subset.
Here's the step-by-step reasoning to find the solution:
1. Start with the Full Set: Consider the set of integers from 1 to 100, i.e., {1, 2, 3, ..., 100}.
2. Initialize an Empty Valid Subset: Begin building a subset where no number is three times another.
3. Iterate Through the List: Consider each number from 1 to 100 one by one.
4. Check the Condition: For each number, check if it is three times any number already in our valid subset or if any number in our valid subset is three times the current number.
5. Build the Subset: If the current number does not violate our condition, add it to the subset. If it does, skip it.
By following this procedure, we end up with a specific set of numbers that adhere to our properties:
- The largest number of members such a subset can have is 76.
- The subset is:
{1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 97, 98, 99, 100}.
By carefully ensuring this condition is met throughout the selection process, we find that the maximum number of members that such a subset can have is indeed 76.
