If a number 45678x9231 is divisible by 3, then how many values are possible for x.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine how many values are possible for \( x \) in the number \( 45678x9231 \) such that it is divisible by 3, we can follow these steps:

### Step 1: Understand the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 2: Calculate the sum of the known digits
First, we need to find the sum of the digits in the number \( 45678x9231 \) without \( x \):

\[ 4 + 5 + 6 + 7 + 8 + 9 + 2 + 3 + 1 = 45 \]

### Step 3: Include \( x \) in the sum
Now, we include \( x \) in the sum:

\[ 45 + x \]

### Step 4: Set up the condition for divisibility by 3
For \( 45678x9231 \) to be divisible by 3, the total sum \( 45 + x \) must be divisible by 3.

### Step 5: Determine the divisibility of 45
Since \( 45 \) is already divisible by 3, we can say:

\[ 45 + x \equiv 0 \mod 3 \]

This simplifies to:

\[ x \equiv 0 \mod 3 \]

### Step 6: Find possible values for \( x \)
The possible values of \( x \) that satisfy \( x \equiv 0 \mod 3 \) and are single-digit numbers (0 to 9) are:

- \( x = 0 \)
- \( x = 3 \)
- \( x = 6 \)
- \( x = 9 \)

### Step 7: Count the possible values
Thus, the possible values for \( x \) are \( 0, 3, 6, \) and \( 9 \). This gives us a total of 4 possible values.

### Final Answer
The number of values possible for \( x \) is **4**.

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