Breaking
Number Theory
Level
3
is a positive integer greater than 5. What is the maximum amount of primes are there in the following sequence?
The answer is 8.
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In the beginning, we start with 30 numbers, and all could be potentially prime.
Every second number is divisible by 2, so we're left with 15 possibilities.
Every third number is divisible by 3 (independently from divisibility by 2), so out of 15 numbers, 10 remain.
Every fifth number is divisible by 5 (independent from previous), so 8 numbers remain.
If a number was divisible by 7 (or a greater prime) then we would have already counted it before.
To prove that it's actually possible to have 8 primes among these numbers, consider n = 6 , which gives the primes 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 , 3 1 .
Therefore, the maximum number of primes is 8 .