If x and y are different prime numbers, each greater than 2, which of the following must be true?

I. x+y is an even integer

II. xy is an odd integer

III. (x/y) is not an integer

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


Kudos for correct solution.
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\(x\) and \(y\) are different prime numbers, each greater than 2.
As 2 is the only even prime number, both \(x\) and \(y\) must be odd.
So, we have three properties that will be useful in solving this question:
1. \(x\) and \(y\) are prime numbers
2. \(x\) and \(y\) are odd numbers
3. \(x\) and \(y\) are different numbers

Now, let us consider each statement.
Statement I: \(x + y\) is an even integer
We know that both \(x\) and \(y\) are odd. We also know that odd + odd = even
hence, \(x + y\) is an even integer.
So, statement I must be true.

Statement II: \(xy\) is an odd integer
We know that both \(x\) and \(y\) are odd. We also know that odd x odd = odd
hence, \(xy\) is an odd integer.
So, statement II must also be true.

Statement III: \(x/y\) is not an integer
This could have been tricky. But we know that \(x\) and \(y\) are different numbers. So, \(x/y\) cannot be 1. We also know that \(x\) and \(y\) are prime numbers. So, \(x\) is definitely not divisible by \(y\).
Hence, \(x/y\) cannot be an integer.
So, statement III must also be true.

Therefore, the correct answer is E.

Hi All,

We're told that a X and Y are DIFFERENT PRIME numbers, each GREATER than 2. We're asked which of the following must be true (which really means "which of these is ALWAYS true no matter how many different examples you can come up with?"). This question can be solved used Number Properties and a bit of logic.

I. X + Y is an EVEN integer

Since X and Y are PRIME numbers AND they're both greater than 2, they must both be ODD numbers.
Odd + Odd = ALWAYS Even, so Roman Numeral 1 is always true.
Eliminate Answers A and D.

II. (X)(Y) is an ODD integer

We already know that X and Y are both ODD numbers.
(Odd)(Odd) = ALWAYS Odd, so Roman Numeral 2 is always true.
Eliminate Answer C.

III. (X/Y) is NOT an integer

For X/Y to be an integer, X must be a MULTIPLE of Y. We're told that X and Y are DIFFERENT numbers - and by definition, a PRIME number has no other factors besides 1 and itself, so it is NOT possible for X to be a multiple of Y. Thus, X/Y will NEVER be an integer under these circumstances and Roman Numeral 3 is always true.
Eliminate Answer B.

Final Answer:

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