CDS 2026 (I) Elementary Mathematics, question 77

Topic: Number system

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Q77

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)(p^{2}+q^{2}) always composite number, where pp and qq are different prime numbers ?

Statement-I : (p−q)(p-q) is an odd integer

Statement-II : (p+q)(p+q) is an odd integer

Which one of the following is correct in respect of the above Question and the Statements ?

  1. (a) The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
  2. (b) The Question can be answered by using either Statement aloneCorrect answer
  3. (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
  4. (d) The Question cannot be answered even by using both the Statements together

Solution.

Answer: (b) The Question can be answered by using either Statement alone

  1. Two different primes have an odd difference only when one of them is 2, because two odd primes give an even difference.
  2. So Statement I alone says one prime is 2 and the other is odd; then p=2p = 2, q=3q = 3 gives p2+q2=13p^{2} + q^{2} = 13, a prime.
  3. Hence the answer to the Question is No, not always, found from Statement I alone.
  4. Statement II (odd sum) gives exactly the same information, so it also answers the Question alone.

Trap: The Statements do not fix the value of p2+q2p^{2} + q^{2}, which makes (d) tempting, but one counterexample is enough to answer an always question with a definite No.