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If x and y are different prime numbers, each greater than 2,

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If x and y are different prime numbers, each greater than 2,

by swerve » Wed Sep 19, 2018 10:14 am

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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

The OA is E.

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If x and y are different prime numbers, each greater than 2,

by regor60 » Wed Sep 19, 2018 10:37 am
swerve wrote: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

The OA is E.

Source: Veritas Prep
Prime numbers are always odd since, if they were even, they would be divisible not just by 1 and itself, but also by 2, which would make them not prime.

An even number can be characterized as 2K where K is any integer. An odd number can be characterized as 2K+1. So let X = 2K+1 and Y = 2M+1

Test the answers:

X+Y = 2K + 2M + 2 = 2(K+M+1). Since this is a multiple of 2, it is even.

XY= (2K+1)(2M+1)= 4KM + 2M + 2K + 1 = 2(2KM+M+K) +1. The first term is even since it is a multiple of 2. Adding 1 to this even number means XY i s odd.

X/Y = (2K+1)/(2M+1) is an integer only if the prime number X is a multiple of Y. But since X is a prime different from Y, it can't be a multiple of Y, so X/Y is not an integer. Answer E
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by Brent@GMATPrepNow » Wed Sep 19, 2018 12:18 pm
swerve wrote: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
Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN

If x and y are different prime numbers, each greater than 2 . . .
Since all prime numbers (EXCEPT 2) are ODD, this statement is telling us that x and y are different ODD numbers

. . . which of the following must be true?

I. x + y is an even integer
x + y = ODD + ODD = EVEN (by rule #1)
This statement is true

II. xy is an odd integer
xy = (ODD)(ODD) = ODD (by rule #4)
This statement is true

III. (x/y) is not an integer
Any prime number is divisible by ONLY 1 and itself
Since x and y are different, and since y cannot equal 1, x/y CANNOT be an integer
This statement is true

Answer: E

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Brent
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by Scott@TargetTestPrep » Thu Sep 27, 2018 4:41 pm
swerve wrote: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
Since x and y must be odd if each is greater than 2, then x + y must be even, and xy must be odd. We see that I and II are true.

Since x and y are different prime numbers, neither can be a multiple of the other, so x/y can't be an integer. III is true also.

Answer: E

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by [email protected] » Thu Sep 27, 2018 6:08 pm
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: E

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Rich
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