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Let be the set of positive integers less than or equal to such that the difference of any two numbers in is neither nor . find the maximum numbers of elements in
Full solution
Q. Let be the set of positive integers less than or equal to such that the difference of any two numbers in is neither nor . find the maximum numbers of elements in
- Pick Smallest Integer: Let's start by picking the smallest positive integer, which is , and add it to set .
- Add Numbers to Set: Now, we can't add or to because the difference between these numbers and is either or . So, the next number we can add is .
- Identify Pattern: Continuing this pattern, we can't add or to , so the next number we can add is .
- Calculate Number of Elements: We notice a pattern: we can add a number to every th integer to avoid differences of or . So, we can calculate the number of elements by dividing by .
- Check Inclusion of Last Numbers: Dividing by gives us with a remainder of . So, we can have elements, and we need to check if we can include the last two numbers, and .
- Verify First and Last Numbers: Since and , we can't include or in . So, we stick with elements.
- Verify First and Last Numbers: Since and , we can't include or in . So, we stick with elements. However, we need to check if the first and last numbers in our set follow the rule. The last number we can include is , but we just determined we can't include .
