The probability that a randomly selected 2-digit number belongs to the set {n ∈ N : (2^n - 2) is a m

Q: Why can't I just calculate $2^n$ for a few numbers and see the pattern?

You could, but that is a trap! $2^n$ grows exponentially, making manual calculation prone to errors. Modular arithmetic is the elegant, bulletproof way to handle divisibility without ever needing to compute the actual value of $2^n$.

Q: Wait, why do we reject the even numbers? Is it because of a negative root?

It is not about roots; it is about parity. If $n$ is even, $(-1)^n$ becomes $1$, and the expression becomes $1 - 2 = -1$. Since $-1$ is not divisible by $3$, even numbers fail the condition. We reject them because they simply do not satisfy the divisibility rule.

Q: Is there a trick to counting the 2-digit numbers?

Always remember the formula: (Last - First) + 1. For 2-digit numbers, that is $99 - 10 + 1 = 90$. Never forget the '+1', or you will be off by one!

Question

The probability that a randomly selected 2-digit number belongs to the set $\{n \in \mathbb{N} : (2^n - 2) 	ext{ is a multiple of } 3\}$ is equal to

Options:

(A) $1/6$

(B) $2/3$

(C) $1/2$

(D) $1/3$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to unravel a beautiful problem that sits at the intersection of number theory and probability.

When you face a problem like this in the JEE Advanced, the first step is always to define your universe. We are looking at two-digit numbers, which range from $10$ to $99$.

To find the total number of outcomes, we use the simple yet vital formula: 
$$n(S) = 99 - 10 + 1 = 90$$ 
This is our sample space, the stage upon which our mathematical drama unfolds.

### The Modular Lens: A Time-Traveler's Tool

Now, let us look at the condition: $(2^n - 2)$ must be a multiple of $3$. In the language of mathematics, we write this as:
$$(2^n - 2) \equiv 0 \pmod{3}$$

When you see large powers like $2^n$, do not reach for a calculator. Reach for modular arithmetic! The secret here is to rewrite the base, $2$, in terms of the modulus, $3$.

We can cleverly write $2$ as $(3 - 1)$. Because $3 \equiv 0 \pmod{3}$, we have $2 \equiv -1 \pmod{3}$. This transforms our expression into:
$$(-1)^n - 2 \equiv 0 \pmod{3}$$

### The Parity Trap: A Fork in the Road

This is where the problem becomes a story of two paths: even numbers and odd numbers. The expression $(-1)^n$ is a chameleon; it changes its value based on the parity of $n$.

If $n$ is even, $(-1)^n$ becomes $1$. Our expression becomes:
$$1 - 2 = -1$$
Since $-1$ is not a multiple of $3$, all even numbers are disqualified from our favorable set.

If $n$ is odd, $(-1)^n$ becomes $-1$. Our expression becomes:
$$-1 - 2 = -3$$
Since $-3$ is perfectly divisible by $3$, every single odd two-digit number is a winner.

### The Victory Lap: Calculating the Probability

We have discovered that our favorable outcomes are exactly the odd two-digit numbers. Since we have $90$ numbers in total, and the sequence alternates between even and odd, exactly half of them are odd.

Thus, the number of favorable outcomes is:
$$n(E) = 45$$

The final step is the easiest part of the journey: calculating the probability. We take our favorable outcomes and divide them by the total sample space:
$$P(E) = \frac{45}{90}$$

Simplifying this fraction gives us the final result:
$$\mathbf{P(E) = \frac{1}{2}}$$

You see? By using the power of modular arithmetic, we turned a daunting problem into a clear, logical path. Keep this mindset, and no JEE problem will ever stand in your way!

Animated Solution for Mathematics - Probability: The probability that a randomly selected 2-digit number belongs to the set ${n \in N : (2^{n} - 2) is a multiple of 3}$ is equal to

Select Answer:

Visualized Solution

Defining the Sample Space

The Condition for Favorable Outcomes

Rewriting the Base

Applying Binomial Expansion

Splitting into Cases

Case 1: When $n$ is Even

Case 2: When $n$ is Odd

Counting Favorable Outcomes

Final Probability Calculation

The Sigma Insight: Classical Definition of Probability

Analyzing the Setup

The Modular Lens

The Parity Trap

The Victory Lap

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Animated Solution for Mathematics - Probability: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that a randomly selected 2-digit number belongs to the set {n∈N:(2n−2) is a multiple of 3} is equal to