If (180! + 1)

Question

If (180! + 1) < P < (180! + 180) and P is a positive integer, then how many prime numbers P can take ?

A. 0
B. 1
C. 2
D. 179
E. 180

chetan2u · Answer

So P is something, say x, added to 180!.
180! is the product of all integers from 1 to 180, and x also lies somewhere between 1 and 180.

We can write P as  180!+x=1*2..x*..179*180+x = x(\frac{180!}{x}+1) 
Thus P will always have a factor x other than 1 and itself.

Thus no prime number exist.

A

Palak74 · Answer

I think the answer is answer option A i.e 0

A factorial can not be prime
And a factorial + a number X (X Being less than or equal to the factorial ) will always be divisible by X

AbRathod · Answer

Can we take a case, similar to smaller value, and still get the right answer?
(3! + 1 < x < 3! + 3) = 7 < x < 9; Answer is eight, which is not a prime number
Answer is Zero (A)

SaquibHGMATWhiz · Answer

Solution:

We know  180!=180	imes 179	imes 178	imes 177 	imes .....	imes 3	imes 2	imes 1  
 Positive integers between  180!+1  and  180!+180  i.e.,  180!+2, 180!+3, 180!+4....180!+177, 180!+178, 180!+179  have some or other other number common between the two terms  
  180!+2 : 2 can be taken out common    180!+3 : 3 can be taken out common    180!+177 : 177 can be taken out common   and so on   Thus none of them are prime

Hence the right answer is    Option A

stne · Answer

I just tested it with a smaller example and it worked!

Let us take a similar even factorial, i.e.  4!

Then  4!+1<  p  < 4!+4

25 <  p  < 28

Hence p can take no prime numbers.

Ans A

MBAApplicant2026 · Answer

If the question instead asked if P was an integer between 180! and 180! + 180, would there be an easy way to tell if 180! + 1 is prime? I understand how 180! + 2, 3, 4, etc are not prime but not sure if it’s just plus 1

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If (180! + 1) < P < (180! + 180) and P is a positive integer, then how

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