Number of four digit positive integers if the product of their digits is divisible by `3` is.

Options

A. `2700`
B. `5464`
C. `6628`
D. `7704`

Explanation

To solve the problem of finding the number of four-digit positive integers whose product of digits is divisible by 3, we can follow these steps:

### Step 1: Calculate the total number of four-digit integers
A four-digit integer ranges from 1000 to 9999. The first digit (thousands place) cannot be 0, so it can be any digit from 1 to 9 (9 options). The other three digits (hundreds, tens, and units) can be any digit from 0 to 9 (10 options each).

Total number of four-digit integers:
\[ \text{Total} = 9 \times 10 \times 10 \times 10 = 9000 \]

### Step 2: Identify the digits that are not divisible by 3
The digits that are divisible by 3 are 0, 3, 6, and 9. Therefore, the digits that are not divisible by 3 are 1, 2, 4, 5, 7, and 8. This gives us 6 options for each digit when we want to ensure that the product of the digits is not divisible by 3.

### Step 3: Calculate the total number of four-digit integers whose product is not divisible by 3
For the first digit (thousands place), we can use any of the 6 digits that are not divisible by 3 (1, 2, 4, 5, 7, 8). For the other three digits, we can also use the same 6 digits.

Total number of four-digit integers whose product is not divisible by 3:
\[ \text{Not Divisible by 3} = 6 \times 6 \times 6 \times 6 = 6^4 = 1296 \]

### Step 4: Calculate the number of four-digit integers whose product is divisible by 3
To find the number of four-digit integers whose product of digits is divisible by 3, we subtract the count of integers whose product is not divisible by 3 from the total count of four-digit integers.

\[ \text{Divisible by 3} = \text{Total} - \text{Not Divisible by 3} \]
\[ \text{Divisible by 3} = 9000 - 1296 = 7704 \]

### Conclusion
The number of four-digit positive integers whose product of digits is divisible by 3 is **7704**.

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