M41-34

Question

If  x  is a positive integer and the product of any two distinct positive factors of  x  is odd, which of the following must be true?
 
I.  x  is an odd prime.
 
II.  x^2  is odd.
 
III.  x  is the square of an odd number.

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

Bunuel · Answer

Official Solution:

If  x  is a positive integer and the product of any two distinct positive factors of  x  is odd, which of the following must be true?
 
I.  x  is an odd prime.
 
II.  x^2  is odd.
 
III.  x  is the square of an odd number.

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

Given that the product of  any  two distinct positive factors of  x  is odd,  x  cannot be even because if it is, then it has 1 and 2 as factors and  1 * 2 = 2 = even . Thus,  x  must be odd. In this case, all its factors will be odd and the product of any two distinct positive factors of  x  will be odd. Therefore, only II must be true because if  x  is odd, then  x^2  will also be odd.

Answer: B

lnyngayan · Answer

So why are statement I and III not true?

Bunuel · Answer

The question asks which of the options MUST be true, meaning which one is ALWAYS true. We deduce that x must be odd. Now, Option I states that x is an odd prime, but this is not necessarily true. Although x must be odd, it doesn't have to be prime; for example, x could be 15, which is not prime. Option III, which states that x is the square of an odd number, is also not necessarily true. Again, we can consider the case when x = 15.

Hope that clarifies things.

rishabh2310 · Answer

why is 3 option wrong , x is the square of an odd number which will also be odd and multiplication of odds will always be odd why cant the answer be option E??? kindly explain thank you in advance

Bunuel · Answer

x is not necessarily the square of an odd number. From the stem, we can only deduce that x is odd—nothing more. Please review the solution again, more carefully this time.

Shabaz18gmat · Answer

I've a doubt @bunel.

question mentioned "    the product of any two distinct positive factors of       x      is odd  "

if x*x is odd, then x is also odd but if we have x=1, then it won't have 2 distinct positive factors.

In my opinion, I satisfies the "must be true" condition as for any odd prime, it has 2 distinct factors, 1 & number itself. Also this option eliminates 1 automatically which was causing trouble because 1 doesn't have 2 distinct positive factors instead it has only 1 factor

Bunuel · Answer

The condition does not require x to have exactly two distinct factors. It says: if you pick any two distinct positive factors, their product is odd.

I is not “must be true” because x can be odd composite, like x = 9, and the condition still holds.

So only II must be true (since the condition forces x to be odd).

luisdicampo · Answer

Deconstructing the Question  
We are told that for the positive integer  x , the product of any two distinct positive factors of  x  is odd.
Key idea: an odd product requires both factors to be odd, so  x  cannot have any even factor.

Step-by-step  
If  x  were even, then  2  would be a positive factor of  x .
Also  1  is a distinct positive factor.
Then  2\cdot 1=2  would be even, contradicting the condition.
So  x  must be odd.

Check the statements:

I.  x  is an odd prime.
Counterexample:  x=9  works (all factors are odd), but  9  is not prime. So I is not must.

II.  x^2  is odd.
Since  x  must be odd,  x^2  must be odd. So II must be true.

III.  x  is the square of an odd number.
Counterexample:  x=15  works (all factors are odd), but  15  is not a square. So III is not must.

Answer: II only

M41-34

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