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I.S.I. and C.M.I. Entrance

Consecutive composites | TOMATO Objective 151

This is an objective problem 151 from TOMATO based on Consecutive composites, useful for Indian Statistical Institute Entrance Exam.

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).

Answer is (A) 

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

By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

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