A 4 digit number is randomly picked from all the 4 digit numbers, then the probability that the product of its digit is divisible by 3 is :

Options

A. `(107)/(125)`
B. `(109)/(125)`
C. `(111)/(125)`
D. None of these

Answer

To find the probability that the product of the digits of a randomly picked 4-digit number is divisible by 3, we can follow these steps:

### Step 1: Determine the total number of 4-digit numbers
A 4-digit number can range from 1000 to 9999. The total number of 4-digit numbers is:
\[ 9999 - 1000 + 1 = 9000 \]

### Step 2: Identify the digits that make the product divisible by 3
The product of the digits of a number is divisible by 3 if at least one of the digits is divisible by 3. The digits that are divisible by 3 from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are:
- 0 (but cannot be the first digit)
- 3
- 6
- 9

### Step 3: Use complementary counting
Instead of directly counting the favorable outcomes, we will count the cases where the product is **not** divisible by 3. This happens when none of the digits are 0, 3, 6, or 9. The remaining digits are:
- 1, 2, 4, 5, 7, 8 (which are 6 digits)

### Step 4: Count the cases where the product is not divisible by 3
1. The first digit (thousands place) cannot be 0, so it can be one of the 6 digits (1, 2, 4, 5, 7, 8).
2. The remaining three digits (hundreds, tens, and units) can be any of the 6 digits (including 0).

Thus, the total number of 4-digit numbers where the product of the digits is not divisible by 3 is:
\[ 6 \times 6 \times 6 \times 6 = 6^4 = 1296 \]

### Step 5: Calculate the probability
Now, we can find the probability that the product of the digits is divisible by 3 using the complementary probability:
\[ P(\text{Product divisible by 3}) = 1 - P(\text{Product not divisible by 3}) \]
\[ P(\text{Product not divisible by 3}) = \frac{1296}{9000} \]
Thus,
\[ P(\text{Product divisible by 3}) = 1 - \frac{1296}{9000} = \frac{9000 - 1296}{9000} = \frac{8704}{9000} \]

### Step 6: Simplify the fraction
To simplify \( \frac{8704}{9000} \):
- The GCD of 8704 and 9000 is 16.
- Dividing both the numerator and denominator by 16 gives:
\[ \frac{8704 \div 16}{9000 \div 16} = \frac{544}{562.5} \text{ (not an integer, so keep it as is)} \]

Thus, the final probability that the product of the digits of a randomly picked 4-digit number is divisible by 3 is:
\[ \frac{8704}{9000} \approx 0.967 \]

### Final Answer:
The probability that the product of the digits of a randomly picked 4-digit number is divisible by 3 is \( \frac{8704}{9000} \).

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