N is an integer and N>2, at most how many integers among N + 2, N + 3, N + 4,…

Answer: At most 2 of the six integers can be prime.

Reason:

• Among any six consecutive integers there are three even numbers. Since N > 2, none of N+2,...,N+7 equals 2, so these three even numbers are composite.

• Among any six consecutive integers there are two multiples of 3. One of those multiples of 3 is even (the one divisible by 6), so at least one odd multiple of 3 is present among the six, and that odd multiple of 3 is composite.

• Counting distinct composite numbers gives at least four composites (the three evens plus at least one odd multiple of 3), so at most 6 - 4 = 2 of the six numbers can be prime.

• This bound is achievable: for example, with N = 3 the six numbers are 5, 6, 7, 8, 9, 10, and the primes are 5 and 7 (two primes).