Animated Solution for Mathematics - Basic Mathematics: Consider the statement: "For an integer n, if $n^3-1$ is even, then n is odd." The contrapositive statement of this statement is :

Welcome, future engineer. Today, we aren't just solving a problem; we are peeling back the curtain on the very language of mathematics: Logic.

Often, students rush to calculate, but in JEE Advanced, the ability to manipulate logical statements is a superpower. Let's look at our statement: "For an integer $n$, if $n^3-1$ is even, then $n$ is odd."

This is a conditional statement, the bread and butter of rigorous proof. We denote this as $P\to Q$. Here, $P$ is the hypothesis: "$n^3-1$ is even", and $Q$ is the conclusion: "$n$ is odd".

Now, the question asks for the contrapositive. Why do we care? Because in the world of logic, $P\to Q$ and its contrapositive $ eg Q \to eg P$ are twins—they are logically equivalent.

To find the contrapositive, we perform a two-step dance: we swap the positions of $P$ and $Q$, and we negate both.

First, we must isolate our components. The hypothesis $P$ is "$n^3-1$ is even". The conclusion $Q$ is "$n$ is odd".

This structure is the foundation. If you misidentify these, the rest of the logic collapses. Always take a moment to write them down clearly.

The contrapositive is defined as $ eg Q \to eg P$. This means we take the conclusion, negate it, and make it the new hypothesis. Then, we take the hypothesis, negate it, and make it the new conclusion.

Let's start with the conclusion $Q$: "$n$ is odd". Negating this, $ eg Q$, gives us "$n$ is not odd".

Since we are dealing with integers, the only alternative to odd is even. So, $ eg Q$ becomes "$n$ is even".

Next, we look at the hypothesis $P$: "$n^3-1$ is even". Negating this, $ eg P$, gives us "$n^3-1$ is not even", which simplifies to "$n^3-1$ is odd".

Now, we assemble our new statement: "If $ eg Q$, then $ eg P$".

Putting it together, we get: "For an integer $n$, if $n$ is even, then $n^3-1$ is odd."

This is the elegance of logic. We didn't need to test values; we simply transformed the structure. Keep this "swap and negate" rule in your toolkit, and you will never fear a logic question again.

You have successfully navigated the logical landscape. Keep practicing, and remember: math is not just about numbers; it is about the structure of truth itself.