For any integer n greater than 1, find the number of prime numbers between n!+1 and n!+n.?

Question

For any integer n greater than 1, find the number of prime numbers between n!+1 and n!+n

Answer 1

The answer is 0. For 2 ≤ x ≤ n and for integer n > 1,

n! + x = x * (n!/x + 1)

which means it will always be divisible by both

x

and

n!/x + 1

To demonstrate, for n = 5

(2 * 3 * 4 * 5) + 2 = 2 * (3 * 4 *5 + 1)
(2 * 3 * 4 * 5) + 3 = 3 * (2 * 4 *5 + 1)
(2 * 3 * 4 * 5) + 4 = 4 * (2 * 3 *5 + 1)

Answer 2

The number is zero.

Maybe n! + 1 can be prime, and n! + n + 1 can be prime. But for all of the other numbers n! + k in between these, there is a factor of k somewhere within the n!, so k is a factor of the number.

Answer 3

Option A is correct. Explanation.. Take any value of N (not a Prime No)=4 thn 4! = 24, therefore 4!- 1 =23 its a prime no... Nxt take ne value of N (Prime No) = 5 thn 5! = 120, therefore 5! - 1= 119 ...Prime No.... So ans A is correct.... None of the other options are justified...

Answer 4

any prime no will be of the form 6k+-1....for k<>0 and all the nos that you chose are typically n!+1,n!+2......and n!+(n-1)...
n! is never going to be prime also we can leave the even no terms since the value of n! is always even for n>1...
I am not sure, i tried until n=6 and found that there are no such nos and hence I assume(?????) that there are no prime nos between n! +1 and n!+n... Sorry if I am wrong..

Answer 5

2, mabe