Three, Five, and Seven have Different Remainders When Divided by Three

There are many cute math problems that use the trivial fact announced in the title. For example, I recently posted the following problem from the 43rd Tournament of Towns.

Problem. Find the largest number n such that for any prime number p greater than 2 and less than n, the difference np is also a prime number.

Solution. Prime numbers 3, 5, and 7 have different remainders modulo 3. Thus, for any n, one of the numbers n − 3, n − 5, or n − 7 is divisible by 3. If n > 10, that number that is divisible by 3 is also greater than 3, thus, making it composite. Therefore, the answer to this problem is not greater than 10. Number 10 works, thus, the answer is 10.

Here is another problem using the fact that 3, 5, and 7 have different remainders when divided by 3.

Problem. Find the maximum integer n, such that for any prime p, where p < n, the number n + 2p is prime.


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