If x is a positive integer and the difference between any two

Question

If  x  is a positive integer and the difference between any two distinct positive factors of  x  is even, which of the following must be true?

I.  x^2 - 1  is even.

II.  \frac{x+1}{2}  is even.

III.  x  is not a prime number.

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Bunuel · Accepted Answer

Official Solution:

If  x  is a positive integer and the difference between any two distinct positive factors of  x  is even, which of the following must be true?

I.  x^2 - 1  is even.

II.  \frac{x+1}{2}  is even.

III.  x  is not a prime number.

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Given that the difference of  any  two distinct positive factors of  x  is even,  x  cannot be even because if it is, then it has 1 and 2 as factors and  1 - 2 = 1 = odd . Thus,  x  must be odd. In this case, all its factors will be odd and the difference of any two distinct positive factors of  x  will be even.

If  x  is odd, then  x^2 - 1 = 	ext{odd}^2 - 1 = 	ext{odd} - 1 = 	ext{even} . Thus, I must be true. To discard the other two options, consider  x = 5 .

Answer: A

gmatophobia · Answer

Inference   :  x is an odd number, as all the factors of x are odd.

Answer choice elimination

I.  x^2 - 1  is even.

We know that x is odd. Hence  x^2  is also odd.

x^2 - 1  = even

This must be true.

II.  \frac{x+1}{2}  is even.

We know that x is odd. Hence, x + 1 = even.

However, if  x+1  is even,  \frac{x + 1}{2}  may or may not be even.

Ex.  x = 5

x + 1 = 6

\frac{x + 1}{2 } = 3   = odd

III.  x  is not a prime number.

x can be a prime number. For example, x can be equal to 5.

Option A

samarpan.g28 · Answer

The difference between the two distinct factors is even, and we know odd-odd=even. So if both the factors are odd then odd*odd=odd so x is definitely an odd number.
I.   x^2 - 1  is even - true
II.  \frac{x+1}{2}  is even - not necessarily true since x can be 5,13,17 etc in such case the result will be 3,7,9 which are odd so false.
III.  x  is not a prime number - x can be 5,13,17 etc. so false.

Option (A) is correct.

Rahuln2501 · Answer

x is positive integer
Since the difference between any two distinct positive factors of x is even,

We know that,
even - even = even 
odd - odd = even
even - odd = odd

If x is even
Factors of x --> 1,2,..something (contains odd and even factors, so the difference between any two could result to odd)

If x is odd
Factors of x ---> 1..something (contains only odd factors, so the difference between any two will always result to even)

So x is odd.

I.  x^2-1  is even ✅

odd ^2 -1 = odd-1 = even

II.  \frac{x+1}{2}  is even ❌

x = 3,  \frac{3+1}{2} = 2   = even

x = 5,  \frac{5+1}{2} = 3   = odd

III. x is not a prime number ❌

x can be 5,7,11,13....
 
 Option (A) is correct.

bumpbot · Answer

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If x is a positive integer and the difference between any two

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If x is a positive integer and the difference between any two

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