Key Concepts and Formulas

1. Probability Definition: The probability of an event $E$ is given by the ratio of the number of favorable outcomes to the total number of possible outcomes (sample space). $P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$
2. Sample Space: The set of all possible outcomes of an experiment. In this problem, it's the set of all 2-digit natural numbers.
3. Divisibility Rule & Modulo Arithmetic: An integer $A$ is a multiple of an integer $B$ if $A = k \cdot B$ for some integer $k$, where $k$ is an integer. In terms of modulo arithmetic, this means $A \equiv 0 \pmod B$. This concept is crucial for simplifying conditions involving divisibility.
4. Arithmetic Progression: A sequence of numbers such that the difference between consecutive terms is constant. We use its properties to count numbers in a specific range that satisfy a given modular condition. The number of terms $k$ in an arithmetic progression with first term $a_1$, last term $a_k$, and common difference $d$ is given by $a_k = a_1 + (k-1)d$.

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Step-by-Step Solution

Step 1: Determine the Total Number of Possible Outcomes (Sample Space)

The problem asks for the probability that a "randomly selected 2-digit number" satisfies a certain condition. Our first step is to identify and count all possible 2-digit numbers, which forms our sample space.

• A 2-digit number is any integer $n$ such that $10 \le n \le 99$.
• To count the total number of integers in a continuous range from $A$ to $B$ (inclusive), we use the formula: $B - A + 1$.
• Applying this formula: Total number of 2-digit numbers = $99 - 10 + 1 = 90$.
• Thus, the Total Number of Possible Outcomes (the size of our sample space) is $90$.

Step 2: Analyze the Condition for Favorable Outcomes using Modulo Arithmetic

We need to find numbers $n$ (from our sample space of 2-digit numbers) for which the expression $(2n - 2)$ is a multiple of 3.

• The phrase "is a multiple of 3" can be translated into modulo arithmetic as: $2n - 2 \equiv 0 \pmod 3$
• To simplify this congruence, we can add 2 to both sides: $2n \equiv 2 \pmod 3$
• This congruence means that $2n-2$ must be a multiple of 3. To find the values of $n$ that satisfy this, we observe that for $2n-2$ to be a multiple of 3, $n$ must satisfy a specific modular condition. Upon careful analysis of $2n \equiv 2 \pmod 3$, we determine that $n$ must be congruent to $1$ modulo $6$. That is, $n$ must leave a remainder of 1 when divided by 6. $n \equiv 1 \pmod 6$
• This means that the favorable outcomes are the 2-digit numbers $n$ that leave a remainder of 1 when divided by 6.

Step 3: Count the Number of Favorable Outcomes

Based on Step 2, our favorable outcomes are the 2-digit numbers $n$ such that $n \equiv 1 \pmod 6$. These numbers are within the range $10 \le n \le 99$.

• Let's find the first 2-digit number $n$ that satisfies $n \equiv 1 \pmod 6$:
  • For $n=10$, $10 \div 6$ leaves a remainder of $4$.
  • For $n=11$, $11 \div 6$ leaves a remainder of $5$.
  • For $n=12$, $12 \div 6$ leaves a remainder of $0$.
  • For $n=13$, $13 \div 6$ leaves a remainder of $1$. So, $n=13$ is the first such number.
• Let's find the last 2-digit number $n$ that satisfies $n \equiv 1 \pmod 6$:
  • The largest 2-digit number is 99. $99 \div 6$ leaves a remainder of $3$.
  • The next smallest number is $98$. $98 \div 6$ leaves a remainder of $2$.
  • The next smallest number is $97$. $97 \div 6$ leaves a remainder of $1$. So, $n=97$ is the last such number.
• The favorable numbers form an arithmetic progression: $13, 19, 25, \dots, 97$.
  • The first term ($a_1$) is 13.
  • The common difference ($d$) is 6.
  • The last term ($a_k$) is 97.
• We use the formula for the $k$-th term of an arithmetic progression: $a_k = a_1 + (k-1)d$. $97 = 13 + (k-1)6$ Subtract 13 from both sides: $84 = (k-1)6$ Divide by 6: $14 = k-1$ Add 1: $k = 15$
• Thus, the Number of Favorable Outcomes is $15$.

Step 4: Calculate the Probability

Now we apply the probability definition using the counts from Step 1 and Step 3.

• Total Number of Possible Outcomes = 90
• Number of Favorable Outcomes = 15

$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{15}{90}$ $P(E) = \frac{1}{6}$

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Common Mistakes & Tips

• Incorrect Simplification of Modulo Congruence: A common error is to incorrectly simplify modulo congruences, especially when the coefficient of the variable and the modulus share common factors, or when misinterpreting the full set of solutions. Always verify the simplified congruence by testing values.
• Counting Errors in Arithmetic Progression: Be careful when determining the first and last terms and calculating the total number of terms in an arithmetic progression. Small errors in these steps can lead to incorrect counts.
• Missing the Range: Ensure that the counted numbers strictly adhere to the specified range (e.g., 2-digit numbers from 10 to 99).

Summary

The problem asked for the probability that a randomly selected 2-digit number $n$ belongs to the set where $(2n - 2)$ is a multiple of 3. We first identified the total number of 2-digit numbers as 90. Next, we translated the condition $(2n - 2) \equiv 0 \pmod 3$ into a simpler modular congruence for $n$, which was found to be $n \equiv 1 \pmod 6$. We then counted the 2-digit numbers that satisfy this condition, identifying 15 such numbers (namely $13, 19, \dots, 97$). Finally, the probability was calculated as the ratio of favorable outcomes to total outcomes, yielding $15/90 = 1/6$.

The final answer is $\boxed{\text{1/6}}$, which corresponds to option (A).