Is the positive integer x even? (1) x is NOT divisible by n!, where n

Question

Is the positive integer x even?

(1) x is NOT divisible by n!, where n is a positive integer.
(2) x^2 is divisible by n!, where n is a positive integer.

M36-68

Bunuel · Accepted Answer

Official Solution:

Is the positive integer  x  even? 
 
(1)  x  is NOT divisible by  n! , where  n  is a positive integer.
 
If  n=2 , then the above would mean that  x  is NOT divisible 2, so in this case  x  must be odd;
 
If  n=3 , then the above would mean that  x  is NOT divisible 6, so in this case  x  could be even (say 2) as well as odd (say 1).
 
Not sufficient.
 
 From (1) we can deduce the following:  n 
eq 1 , because every number is divisible by 1.  
 
(2)  x^2  is divisible by  n! , where  n  is a positive integer.
 
If  n > 1 , then  n!  is even and if  x^2  is divisible by an even number, then that would mean that  x  itself must be even;
 
If  n=1 , then  x  could be any number, even or odd.
 
Not sufficient.
 
(1)+(2) Since (1) ruled out  n=1  case then from (2) we are left with  n > 1  case, which means that  x  is even. Sufficient.

Answer: C

Kinshook · Answer

Q: Is the positive integer x even?

S1: 
x is NOT divisible by n!, where n is a positive integer.
There is no information about n.
NOT SUFFICIENT

S2:
x^2 is divisible by n!, where n is a positive integer
There is no information about n.
NOT SUFFICIENT

S1&S2:
S1: x is NOT divisible by n!, where n is a positive integer.
S2: x^2 is divisible by n!, where n is a positive integer
Still there is no information about n. 
NOT SUFFICIENT.

IMO E

Bunuel · Answer

The question is NOT flawed. The statement can be true simultaneously. For example, x = 12 and n = 4.

Kinshook · Answer

Corrected.

vinayakvaish · Answer

N! can be either 1 or greater than 1 in which case it is always even.

Statement 1) X can be smaller than n!, even, odd lot of possibilities
Statement 2) x2 divisible by n! means either x is odd and n! is 1 or x2 is even and n! is greater than 1

Had n! been 1 it would have divided x since x is an integer but n! is greater than 1 that means integer value is 2 or higher hence even. we know that even value cannot divide odd value hence x2 must be even. if x2 is even then x must be even.

Hence C IMO

RupamPaul13 · Answer

Statement 1: X can be odd or even and n can't take value as 1. : Not Sufficient

Statement 2: Scenario 1: X odd and N=1, Statement 2 valid
 Scenario 2: X even and N>1, For all values of N>1 X has to be an even number for X^2 to be divisible by N!.
 Statement 2 alone is insufficient.

Combining both the statements we get X as even and N>1. IMO C( Both sufficient)

Archit3110 · Answer

#1
x is NOT divisible by n!, where n is a positive integer
x can be even or odd in this case ; test with x= 3 and n=2 and x=10 and n=3
#2
x^2 is divisible by n!, where n is a positive integer
x has to be even in this case r even ; x= 6, n= 3 , x can be odd as well at n=1 insufficient
from 1 & 2 
n being 1 then both the cases x is even
IMO C

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Is the positive integer x even? (1) x is NOT divisible by n!, where n

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Is the positive integer x even? (1) x is NOT divisible by n!, where n

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