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

This can also be solved using modular arithmetic (parity) instead of the odd/even wording.
Since X and Y are primes greater than 2, both are odd, so in mod 2 terms: $X \equiv 1 \pmod 2$ and $Y \equiv 1 \pmod 2$.
Adding these: $X + Y \equiv 1 + 1 \equiv 0 \pmod 2$, which means X + Y is always even, for any such X and Y.
Now check 87: $87 \equiv 1 \pmod 2$, so 87 is odd.
Since X + Y is always even and 87 is odd, an even number can never equal an odd number, so $X + Y \neq 87$ always holds true, no matter which two distinct odd primes are chosen.
Checking the difference similarly: $X - Y \equiv 1 - 1 \equiv 0 \pmod 2$, so X - Y is always even, which rules out X - Y = 23 (odd) completely; that statement is never true.
So only the statement $X + Y \neq 87$ is guaranteed for every valid pair.\[\boxed{2}\]