Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements: 1. $(p-r)^{2^{qs}}$ is even. 2. $(q-s)^{q^{2s}}$ is even. 3. $(q + r)^{2^{p + s}}$ is odd. Which of the statements given above are correct?

Question

CSATHardPrelims 2024Maths

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

Which of the statements given above are correct?

Explanation

We are given that:

Let's evaluate each statement:

Statement 1: $(p - r)^2 (q * s)$ is even.

Since $p$ is odd and $r$ is even, $(p - r)$ is odd. Squaring an odd number gives an odd result.
$q$ is odd and $s$ is even, so $q * s$ is even.
An odd number times an even number is even, so this statement is true.

Statement 2: $(q - s) * q^2 * s$ is even.

$q$ is odd and $s$ is even, so $q - s$ is odd.
$q^2$ is odd (since odd * odd = odd), and $s$ is even.
An odd number multiplied by an odd number and then by an even number is even, so this statement is true.

Statement 3: $(q + r)^2 (p + s)$ is odd.

$q$ is odd and $r$ is even, so $(q + r)$ is odd (odd + even = odd).
$p$ is odd and $s$ is even, so $(p + s)$ is odd (odd + even = odd).
An odd number squared gives an odd result, and multiplying two odd numbers gives an odd result. Therefore, this statement is true.