Question

Let N be a positive integer not equal to 1. Then, none of the numbers 2, 3, ..., N is a divisor of (N! - 1). Thus, 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 statement is necessarily correct.

Answer 1

The correct option is D none of the foregoing statement is necessarily correct.

Let's check it option by option.
Taking N=2, we get (N!-1) =(2!-1)=1
which is not divisible by 2 but 1 is not a prime number
Hence, statement (a) is incorrect.
Moving on to option / statement (b), let's consider N=2 again, we see (N!-1)=(2!-1)=1 is not divisible by $2,3,4,\dots 0$. At 0, the expression $\frac{(N!-1)}{(N!-2)}$
becomes undefined.
Hence, statement (b) is not neccessarily true.
Similarly statement (c) is not necessarily true for N =2
Hence, option (d) is correct