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Consecutive composites (TOMATO Objective 151)

Let n = 51! + 1. Then the number of primes among n+1, n+2, … , n+50 is

(A) 0;

(B) 1;

(C) 2;

(D) more than 2;

Discussion:

51! is divisible by 2, 3,… 51.

Hence 51! +2 is divisible by 2, … , 51! + k is divisible by k if $$k \le 51$$

Therefore all of these numbers are composite (none of them are primes).

Note: The above problem can be further extended to say that for any natural number of n, we can have n consecutive composite numbers.

April 22, 2015