Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

TOMATO Objective 153 | ISI Entrance | N! -1

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

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com