Two natrual numbers are randomly chosen and multiplied, then the chance that their product is divisible by 3 is

Options

A. `(4)/(9)`
B. `(5)/(9)`
C. `(2)/(3)`
D. `(1)/(9)`

Answer

To solve the problem of finding the probability that the product of two randomly chosen natural numbers is divisible by 3, we can follow these steps:

### Step-by-Step Solution

1. **Define the Natural Numbers**: Let the two natural numbers be denoted as \( A \) and \( B \).

2. **Define the Product**: The product of these two numbers can be represented as \( P = A \times B \).

3. **Condition for Divisibility by 3**: For the product \( P \) to be divisible by 3, at least one of the numbers \( A \) or \( B \) must be a multiple of 3. This can be expressed as:
\[ P \text{ is divisible by } 3 \text{ if } A \text{ is a multiple of } 3 \text{ or } B \text{ is a multiple of } 3. \]

4. **Calculate the Probability of Non-Divisibility**: Instead of directly calculating the probability that \( P \) is divisible by 3, we can calculate the probability that \( P \) is **not** divisible by 3, and then subtract this from 1. The product \( P \) is not divisible by 3 if both \( A \) and \( B \) are not multiples of 3.

5. **Determine the Probability for One Number**: Consider the first 9 natural numbers (1 to 9):
- The multiples of 3 in this range are 3, 6, and 9.
- Therefore, the numbers that are **not** multiples of 3 are 1, 2, 4, 5, 7, and 8 (total of 6 numbers).

The probability that a randomly chosen number \( A \) is not a multiple of 3 is:
\[ P(A \text{ is not a multiple of } 3) = \frac{6}{9} = \frac{2}{3}. \]

6. **Calculate the Combined Probability**: Since the selection of \( A \) and \( B \) is independent, the probability that both \( A \) and \( B \) are not multiples of 3 is:
\[ P(A \text{ is not a multiple of } 3 \text{ and } B \text{ is not a multiple of } 3) = P(A \text{ is not a multiple of } 3) \times P(B \text{ is not a multiple of } 3 = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9}. \]

7. **Final Probability Calculation**: Now, we can find the probability that the product \( P \) is divisible by 3:
\[ P(P \text{ is divisible by } 3) = 1 - P(P \text{ is not divisible by } 3) = 1 - \frac{4}{9} = \frac{5}{9}. \]

### Conclusion
Thus, the probability that the product of two randomly chosen natural numbers is divisible by 3 is:
\[ \boxed{\frac{5}{9}}. \]