Is positive integer x odd?

(1) The product of any two distinct positive factors of x is odd.
odd x odd = odd
odd x even = even
even x even = even

Therefore all the factors of x are odd.
3x1 = 3
3x5 = 15
3x7 = 21

Therefore x is odd and Statement 1 alone is sufficient.

(2) The difference of any two distinct positive factors of x is even.
even - even = even
even - odd = odd
odd - odd = even

Either all factors are even or all factors are odd.
1, which is odd, is one of the factor.

Therefore all factors of x are odd.
Therefore x is odd and Statement 2 alone is sufficient.

Therefore answer is D.

Official Solution:


Is positive integer \(x\) odd?

(1) 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. Sufficient.

(2) 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. Sufficient.


Answer: D
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Is positive integer x odd?

(1) The product of any two distinct positive factors of x is odd.
This statement says all the factors of x are odd. Since all factors of x are odd, we know that x is odd.
Option I - SUFFICIENT

(2) The difference of any two distinct positive factors of x is even.
The difference of any 2 numbers is odd when either both the numbers are Odd or both Even
Since x is positive integer, we know that 1 is a factor of x.
Now we know that all the factors of x is odd, which means x=odd

Option II - SUFFICIENT

Ans D
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(1) Let, the factors 1*3*5=15;
1*3=3
3*5=15

Sufficient.

(2) Let, the factors 1*3*5=15

3-1=2 even

5-1=4 even

Sufficient.

The answer is D

Stat1: The product of any two distinct positive factors of x is odd.
For getting product as odd, any two distinct factors are odd, so x must be odd. Sufficient.

Stat2: The difference of any two distinct positive factors of x is even.
For getting difference as even, any two distinct factors are odd, so x must be odd. Sufficient.

So, I think D.

Is positive integer x odd?

(1) The product of any two distinct positive factors of x is odd.
The product is odd implies that all factors are odd.
Sufficient

(2) The difference of any two distinct positive factors of x is even.
x-1 also should be even, which gives x as odd.
Sufficient.

Option D

Posted from my mobile device

Great question....


S1: in order for any number to be ODD, it must be “made up” of ONLY ODD Prime Factors and must not have any Prime Factor of 2 in its prime factorization.

If we can take ANY 2 distinct factors and multiply them and get an ODD Result, that means there is NO Prime Factor of 2 that makes up X that can be used to “create/combine” with other Prime Factors to create distinct factors.

Every Prime Factor that makes up X’s Prime Factorization must be ODD in order to have ONLY ODD Factors.

Afterall, if there was just one Prime Factor of 2, then X would be EVEN

S1 Sufficient


S2: the difference between any 2 Distinct positive factors of X is EVEN

odd - odd = even

even - even = even


However, 1 is a factor of every Integer. So even if X was composed of only the Prime Factor of 2, you would still have the one subtraction where: (Even) - 1 = ODD

Again, the only way this can be possible is if every distinct factor of X is ODD, which means X’s prime factorization must be made up of only ODD Prime Factors


D.

Each statement sufficient alone

Posted from my mobile device

Since it is mentioned ANY two in the question, I thought we could have any number of factor so incase 2 factors are odd, then a third number can be even.

Isn't there a language problem here?

Posted from my mobile device

"Any two" means that you could picky any 2 factors, i.e, all possible pairs of factors would result in the mentioned conditioned. It basically means that it doesn't really matter which 2 you pick, the result will satisfy the condition.

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