Understanding the Question

The question asks if the expression \(p^2 + q^2\) is always a composite number, given that \(p\) and \(q\) are two different prime numbers.

A composite number is a positive integer greater than 1 that has factors other than 1 and itself. A prime number is a positive integer greater than 1 with only two factors: 1 and itself.

We need to determine if the given statements are sufficient to answer this question definitively.

Statement I Analysis

Statement I states that \((p - q)\) is an odd integer.

• For the difference of two integers to be odd, one must be even and the other must be odd.
• Since \(p\) and \(q\) are prime numbers, the only even prime is 2. Therefore, one of the primes must be 2, and the other must be an odd prime. Let \(p=2\) and \(q=k\), where \(k\) is an odd prime.
• The expression becomes \(p^2 + q^2 = 2^2 + k^2 = 4 + k^2\).
• We need to check if \(4 + k^2\) is *always* composite for any odd prime \(k\).
• Consider \(k=3\). Then \(4 + 3^2 = 4 + 9 = 13\). 13 is a prime number.
• Consider \(k=11\). Then \(4 + 11^2 = 4 + 121 = 125\). 125 is a composite number (\(125 = 5^3\)).
• Since \(4 + k^2\) can be prime (for \(k=3\)) or composite (for \(k=11\)), Statement I does not allow us to conclude that \(p^2+q^2\) is *always* composite.
• Therefore, Statement I alone is insufficient.

Statement II Analysis

Statement II states that \((p + q)\) is an odd integer.

• Similar to Statement I, for the sum of two distinct primes to be odd, one prime must be 2 (even) and the other must be an odd prime. Let \(p=2\) and \(q=k\), where \(k\) is an odd prime.
• The expression is \(p^2 + q^2 = 2^2 + k^2 = 4 + k^2\).
• As analyzed in the evaluation of Statement I, \(4 + k^2\) can result in a prime number (e.g., 13 when \(k=3\)) or a composite number (e.g., 125 when \(k=11\)).
• Therefore, Statement II alone is insufficient to determine if \(p^2+q^2\) is always composite.

Combined Statements Analysis

Both Statement I (\(p - q\) is odd) and Statement II (\(p + q\) is odd) imply the same condition: one of the primes must be 2, and the other must be an odd prime \(k\).

• The expression under consideration is \(p^2 + q^2 = 4 + k^2\).
• As established in the individual analyses, this expression can yield a prime result (like 13) or a composite result (like 125).
• Using both statements together does not provide additional information or constraints beyond what each statement provides individually.
• Therefore, even combined, the statements are insufficient to definitively answer whether \(p^2+q^2\) is *always* composite.

Conclusion

Since neither Statement I alone, nor Statement II alone, nor both statements together are sufficient to answer the question, the question cannot be answered even by using both statements together.