What is the probability that an integer x $(1 \leq x \leq 25)$ chosen at random is divisible by both 2 and 3?

1. Analyze the question

We are looking for the probability that a randomly selected integer $x$ from the set $\{1, 2, 3, \dots, 25\}$ is divisible by both 2 and 3.

2. Explain the concept

• Divisibility Rule: If a number is divisible by both 2 and 3, it must be divisible by their Least Common Multiple (LCM). The LCM of 2 and 3 is $2 \times 3 = 6$. Therefore, we are looking for multiples of 6 within the range 1 to 25.
• Probability Formula: The probability of an event is given by: $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

3. Show the solution

• Step 1: Identify the total number of outcomes. The integers are from 1 to 25, so there are 25 possible outcomes.
• Step 2: Identify the favorable outcomes. We need to list the multiples of 6 that fall between 1 and 25: • $6 \times 1 = 6$ • $6 \times 2 = 12$ • $6 \times 3 = 18$ • $6 \times 4 = 24$ • ($6 \times 5 = 30$, which is greater than 25, so we stop here). The favorable outcomes are $\{6, 12, 18, 24\}$. There are 4 favorable outcomes.
• Step 3: Calculate the probability. $$\text{Probability} = \frac{4}{25}$$

4. Identify the answer

Comparing our result to the given options: A. $\frac{1}{25}$ B. $\frac{1}{5}$ C. $\frac{4}{25}$ D. $\frac{3}{4}$

The correct option is C.

5. Tip

When a question asks for divisibility by two different prime numbers (like 2 and 3), always find the LCM first. If the numbers were not prime (e.g., 4 and 6), the LCM would be 12, not 24. Always remember: "Divisible by A and B" means "Divisible by LCM(A, B)."