Question

Two different prime numbers X and Y, both are greater than 2, then which of the following must be true?

• A. \(X - Y = 23\)
• B. \(X + Y \neq 87\)
• C. Both 1 and 2
• D. None of the above

Hint:

Every prime greater than 2 is odd, and odd + odd always gives an even number, never an odd number like 87.

Answer 1

The Correct Option is B

Step 1: Note the nature of primes greater than 2.
The only even prime number is 2. Every prime number greater than 2 is odd. So X and Y are both odd numbers.
Step 2: Test option 1, X - Y = 23.
The difference of two odd numbers is always even (odd minus odd = even). 23 is odd, so X - Y can never equal 23. This statement is never true, hence it cannot be a "must be true" statement.
Step 3: Test option 2, X + Y \(\neq\) 87.
The sum of two odd numbers is always even (odd plus odd = even). 87 is odd, so X + Y can never equal 87, for any choice of odd primes X and Y. This means X + Y \(\neq\) 87 is true for every possible pair, so it must always be true.
Step 4: Test option 3, both 1 and 2.
Since statement 1 is never true, the combination "both 1 and 2" is false.
Step 5: Test option 4, none of the above.
Since statement 2 is always true, this option is also wrong.
Step 6: Conclusion.
Only statement 2 always holds for every pair of distinct odd primes, so option 2 is correct.