CDS 2026 · PYQ · Data Sufficiency / Number Theory · hard
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is (p² + q²) always composite number, where p and q are different prime numbers?
Statement-I: (p - q) is an odd integer
Statement-II: (p + q) is an odd integer
Which one of the following is correct in respect of the above Question and the Statements?
A.The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
B.The Question can be answered by using either Statement alone✓ Correct
C.The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
D.The Question cannot be answered even by using both the Statements together
Explanation
If p and q are different primes and either (p - q) or (p + q) is odd, then exactly one of p, q must be even, which means one of them is 2 (the only even prime). Without loss of generality let p = 2, then p² + q² = 4 + q² where q is an odd prime. Then q² is odd, so 4 + q² is odd. Also q² ≡ 1 (mod 4) for any odd prime q, so 4 + q² ≡ 5 (mod 4) ≡ 1 (mod 4). More importantly, 4 + q² > 5 and we can check: if q = 3, 4+9=13 (prime, not composite); if q = 5, 4+25=29 (prime); if q = 7, 4+49=53 (prime). So the answer to the question is NO — not always composite. Both statements independently lead to the same conclusion (one prime is 2), so either statement alone answers the question (with answer 'No').
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