Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements:

Correct Option (4)

Given that p and q are odd positive integers, and r and s are even positive integers. We evaluate each statement based on the properties of odd and even numbers:

• Statement 1: (p – r)² (qs)

  • The term (p – r) is the difference between an odd integer and an even integer, which results in an odd integer. Thus, (p – r) = odd.
  • Squaring an odd integer yields an odd integer: (p – r)² = (odd)² = odd.
  • The product (qs) involves an odd integer multiplied by an even integer, which results in an even integer: (qs) = odd × even = even.
  • Therefore, the expression (p – r)² (qs) = odd × even = even. Statement 1 is correct.

• Statement 2: (q – s) q² s

  • The term (q – s) is the difference between an odd integer and an even integer, resulting in an odd integer: (q – s) = odd.
  • Squaring an odd integer yields an odd integer: q² = (odd)² = odd.
  • The term 's' is an even integer.
  • Therefore, the expression (q – s) q² s = odd × odd × even = even. Statement 2 is correct.

• Statement 3: (q + r)² (p + s)

  • The term (q + r) is the sum of an odd integer and an even integer, which results in an odd integer: (q + r) = odd.
  • Squaring an odd integer yields an odd integer: (q + r)² = (odd)² = odd.
  • The term (p + s) is the sum of an odd integer and an even integer, which results in an odd integer: (p + s) = odd.
  • Therefore, the expression (q + r)² (p + s) = odd × odd = odd. Statement 3 is correct.

Since all three statements (1, 2, and 3) are correct, option 4 is the correct answer.

Incorrect Options:

Options 1, 2, and 3 are incorrect because they do not include all the statements that have been determined to be correct. Specifically:

• Option 1 (1 and 2 only) is incorrect as Statement 3 is also correct.
• Option 2 (2 and 3 only) is incorrect as Statement 1 is also correct.
• Option 3 (1 and 3 only) is incorrect as Statement 2 is also correct.