N! -1 (TOMATO Objective 153)

Let N be a positive integer not equal to 1. Then note that none of the numbers 2, 3, … , N is a divisor of (N! -1). From this we can conclude that:

(A) (N! – 1) is a prime number;

(B) at least one of the numbers N+1 , N+2 , …., N! – 2 is a divisor of (N! -1);

(C) the smallest number between N and N! which is a divisor of (N!-1), is a prime number;

(D) none of the foregoing statements is necessarily correct;

Discussion:

N! – 1 could be a prime (example N = 4 gives N! – 1 = 23). It might not be a prime as well (example N = 5 gives N! – 1 = 119 which is divisible by 7).

Hence none of option A or B is necessarily true.

However option (C) is true as N! – 1 is not a prime, it’s first prime divisor occurs between N and N! (since none of the primes from 1 to N divides it. Even if it is a prime, the smallest number between N and N! that divides it is itself which is a prime.

Answer: C

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