Four numbers are multiplied together. Then the probability that the product will be divisible by 5 or 10 is :

Options

A. a. `169/625`
B. b. `369/625`
C. c. `169/1626`
D. d. none of these

Answer

To solve the problem of finding the probability that the product of four numbers is divisible by 5 or 10, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the probability that the product of four randomly chosen digits (from 0 to 9) is divisible by either 5 or 10.

2. **Identifying Divisibility Conditions**:
- A number is divisible by 5 if its last digit is either 0 or 5.
- A number is divisible by 10 if its last digit is 0.

3. **Total Possible Outcomes**:
Each of the four numbers can be any digit from 0 to 9. Therefore, the total number of combinations of four digits is:
\[ 10^4 = 10000 \]

4. **Finding Non-Divisible Outcomes**:
To find the probability that the product is **not** divisible by 5 or 10, we need to consider the digits that do not contribute to divisibility:
- The digits that do not lead to divisibility by 5 or 10 are 1, 2, 3, 4, 6, 7, 8, and 9 (total of 8 digits).
- Thus, the number of ways to choose four digits that are not 0 or 5 is:
\[ 8^4 = 4096 \]

5. **Calculating the Probability**:
The probability that the product is not divisible by 5 or 10 is given by the ratio of non-divisible outcomes to total outcomes:
\[ P(\text{not divisible by 5 or 10}) = \frac{8^4}{10^4} = \frac{4096}{10000} \]

6. **Finding the Required Probability**:
The probability that the product is divisible by 5 or 10 is the complement of the above probability:
\[ P(\text{divisible by 5 or 10}) = 1 - P(\text{not divisible by 5 or 10}) = 1 - \frac{4096}{10000} \]
\[ = \frac{10000 - 4096}{10000} = \frac{5904}{10000} = \frac{369}{625} \]

### Final Answer:
The probability that the product of the four numbers is divisible by 5 or 10 is:
\[ \frac{369}{625} \]