Determine the largest number of distinct positive integers you can have such that the difference between any two of them is prime, and this difference divides both of those numbers.
If there are at least 2 odd numbers, then their difference is even, but an even number cannot divide an odd number. Therefore, there is at most 1 odd number. If there are at least 3 even numbers, then at least 2 of them have to differ by an even number at least 4, but that cannot be prime. Therefore, there are at most 2 even numbers. Then, there are at most 3 numbers.
It is possible to do it with 3 numbers. For example, 10, 12, and 15 work. 12-10=2, which divides both 10 and 12. 15-10=5, which divides both 10 and 15. 15-12=3, which divides both 12 and 15. Therefore, the most numbers possible is 3.
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Posted by Math Man
on 2024-03-10 15:02:02 |
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