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

Question

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

I. x + y ≠ 91 
II. x - y is an even 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

CounterSniper · Accepted Answer

I. x + y ≠ 91

Prime numbers say p > 2 are always odd
odd + odd = even 
since 91 is odd 
This statament will always be true

II. x - y is an even integer 
odd - odd = even 
This statement will be always true

III. x/y is not an integer 
since x and y both are prime numbers , y will never evenly divide x 
This statement will be always true

E

Akash720 · Answer

Prime number greater than 2 are all odd integers. Hence, we need to apply even-odd property here. Not to miss the statement given, both are different prime numbers.

I. Odd+Odd is always even - True
II. Odd - Odd is always even - True
III. Prime/Prime can never be an integer as factors of a prime are 1 and itself - True

+1 for E

abhishekdadarwal2009 · Answer

1. prime numbers other than 2 are all odd,
then odd+odd is always=even number, so x+y can never be =91.

2.prime numbers are odd always(other than 2) odd-odd=even always

3.prime numbers are divisible by itself or by 1, so x/y is never an integer.

All are always true. option E

EgmatQuantExpert · Answer

Solution 
  Given:  
 • x and y are two different prime numbers, each greater than 2

To find:  
 • Which of the given statements is/are always true?

Approach and Working:   
Any prime number, which is greater than 2, is odd
 • Thus, x and y both are odd, which implies, x + y = even
 o Thus, Statement 1, “x + y ≠ 91”, is true 
• x and y both are odd, which implies, x - y = even
 o Thus, Statement 2, “x - y is an even integer”, is true 
• Since, x and y are distinct prime numbers, they do not have any common factor except 1.
 o Thus, Statement 3, “x/y is not an integer”, is true

Hence, the correct answer is option E.

Answer: E

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-NP-e1533292073470.jpg

ScottTargetTestPrep · Answer

Since x and y must be odd if each is greater than 2, so both x + y and x - y must be even. We see that I and II are true.

Since x and 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

HITMAN09 · Answer

Given 
X and Y are prime number greater than 2, resulting both these numbers are Old Integer

Concept :- Odd +/- Odd = Even

St1 x + y ≠ 91.

By rephrasing, this statement becomes :- Odd + Odd is not equal to Odd. Based on the provided information, this statement is true
 
St2. x - y is an even integer 
odd - odd = even. Based on the provided information, this statement is true

St3. x/y is not an integer 
Since A prime number is a whole number greater than 1 ( in our case 2) whose only factors are 1 and itself, and x and y both are prime numbers, y is not the factor of X. Hence, this statement is true.

ShikharJain22 · Answer

For statement II
Lets say, x=3 and y=5
then, x-y = -2

Hence, -2 is not an even integer, its just a negative integer
As for negative integers, the O/E concept doesn't hold!!!

ShikharJain22 · Answer

For statement II
Lets say, x=3 and y=5
then, x-y = -2

Hence, -2 is not an even integer, its just a negative integer
As for negative integers, the O/E concept doesn't hold!!!

egmat · Answer

This is a great question that tests your understanding of prime number properties and logical reasoning. Let me walk you through how to approach this systematically.

Here's the key insight that unlocks this problem:

Notice that all prime numbers greater than 2 must be odd. Why? Because if a number greater than 2 is even, it would be divisible by 2, which means it can't be prime. So both  x  and  y  are odd numbers – this is the foundation for analyzing all three statements.

Let's evaluate each statement:

Statement I:  x + y 
eq 91

When you add two odd numbers together, what do you get? Let's think about this:
 
  3 + 5 = 8  (even) 
  7 + 11 = 18  (even) 
  5 + 13 = 18  (even)

In general, odd + odd = even. This is because any odd number can be written as  2k + 1 , so  (2j + 1) + (2k + 1) = 2(j + k + 1) , which is even.

Now, is 91 even or odd? Well,  91 = 90 + 1 , so 91 is odd.

Since  x + y  must always be even (sum of two odd numbers), and 91 is odd, we can never have  x + y = 91 . Therefore, Statement I must be true.

Statement II:  x - y  is an even integer

When you subtract one odd number from another, what happens?
 
  11 - 5 = 6  (even) 
  13 - 7 = 6  (even)

In general, odd - odd = even, because  (2j + 1) - (2k + 1) = 2(j - k) , which is even.

So Statement II must always be true.

Statement III:  x/y  is not an integer

For  x/y  to be an integer,  x  would need to be divisible by  y . But here's the thing – since both  x  and  y  are different prime numbers, neither can be divisible by the other.

Why? If  x  were divisible by  y , then  y  would be a factor of  x . But  x  is prime, so its only factors are 1 and  x  itself. Since  y > 2  and  y 
eq x ,  y  cannot be a factor of  x .

Therefore,  x/y  can never be an integer, so Statement III must be true.

Conclusion:

All three statements (I, II, and III) must be true for any two different prime numbers greater than 2.

Answer: (E) I, II, and III

Want to master the complete framework?

While this explanation covers the core logic, the complete solution on Neuron shows you the systematic approach to tackle all "must be true" problems, common traps students fall into (like confusing "must be true" vs "could be true"), and how to verify your reasoning efficiently. You can check out the  detailed solution on Neuron by e-GMAT  to understand the broader patterns and process skills that apply to similar questions. You can also explore comprehensive solutions for  other GMAT official questions on Neuron  with practice quizzes and detailed analytics to strengthen your foundation.

Hope this helps! 🎯

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