If n positive integers are taken at random and multiplied together, then the probability that the last digit of the product is 2,4,6 or 8, is

Options

A. `(8^(n))/(5^(n))`
B. `(8^(n)-2^(n))/(5^(n))`
C. `(4^(n)-2^(n))/(5^(n))`
D. none of these

Answer

To solve the problem of finding the probability that the last digit of the product of \( n \) positive integers is 2, 4, 6, or 8, we can follow these steps:

### Step 1: Understand the last digit of products
The last digit of a product depends on the last digits of the numbers being multiplied. The possible last digits of positive integers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

### Step 2: Identify favorable outcomes
We are interested in the last digits 2, 4, 6, and 8. These digits are even, which means we need to consider the last digits of the integers that contribute to these outcomes.

### Step 3: Count the total possible last digits
The total last digits of positive integers from 1 to 10 are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

### Step 4: Identify the last digits that lead to even outcomes
The last digits that can lead to an even product (2, 4, 6, 8) are:
- 2, 4, 6, 8 (favorable outcomes)
- 0 (not favorable for our case since we want only 2, 4, 6, or 8)

### Step 5: Calculate the probability of getting an even last digit
The even last digits from 0 to 9 are:
- 0, 2, 4, 6, 8

Thus, there are 5 even last digits. Out of these, 4 are favorable (2, 4, 6, 8).

### Step 6: Calculate the probability
The probability \( P \) that the last digit of the product is 2, 4, 6, or 8 can be calculated as follows:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{10} = \frac{2}{5} \]

### Step 7: Consider the number of integers \( n \)
Since we are taking \( n \) positive integers, the probability remains the same for each integer chosen. Therefore, the probability that the last digit of the product of \( n \) integers is 2, 4, 6, or 8 is:

\[ P = \left(\frac{2}{5}\right)^n \]

### Final Result
Thus, the probability that the last digit of the product of \( n \) positive integers is 2, 4, 6, or 8 is:

\[ \frac{2}{5}^n \]

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