The smallest digit which can replace `**` to make 201 `**` 58 divisible by 9 is

Options

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

Answer

To determine the smallest digit that can replace `**` in the number 201 `**` 58 to make it divisible by 9, we will follow these steps:

### Step 1: Understand the Rule for Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Identify the Digits in the Number
The number we are working with is 201 `**` 58. The digits in this number are:
- 2
- 0
- 1
- `*` (the digit we need to find)
- 5
- 8

### Step 3: Calculate the Sum of the Known Digits
Now, we will calculate the sum of the known digits:
\[ 2 + 0 + 1 + 5 + 8 = 16 \]

### Step 4: Include the Unknown Digit
Let the unknown digit be represented as `x`. Therefore, the total sum of the digits becomes:
\[ 16 + x \]

### Step 5: Set Up the Condition for Divisibility by 9
For the number to be divisible by 9, the sum \( 16 + x \) must be divisible by 9. We can express this as:
\[ 16 + x \equiv 0 \, (\text{mod} \, 9) \]

### Step 6: Calculate the Remainder of 16 When Divided by 9
Calculating \( 16 \mod 9 \):
\[ 16 \div 9 = 1 \quad \text{(remainder 7)} \]
Thus,
\[ 16 \equiv 7 \, (\text{mod} \, 9) \]

### Step 7: Solve for `x`
To satisfy the condition \( 16 + x \equiv 0 \, (\text{mod} \, 9) \), we need:
\[ 7 + x \equiv 0 \, (\text{mod} \, 9) \]
This means:
\[ x \equiv -7 \equiv 2 \, (\text{mod} \, 9) \]

### Step 8: Find the Smallest Non-Negative Digit
The smallest non-negative digit that satisfies \( x \equiv 2 \, (\text{mod} \, 9) \) is:
\[ x = 2 \]

### Conclusion
Thus, the smallest digit that can replace `**` to make 201 `**` 58 divisible by 9 is **2**.

---