What least number should be replaced for `**` so that the number `67301 ** 2` is exactly divisible by `9` ?

Options

A. `5`
B. `6`
C. `7`
D. `8`

Answer

To determine the least number that should replace `**` in `67301**2` so that the entire number is exactly divisible by `9`, we can 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: Write Down the Known Digits
The digits of the number `67301**2` are `6`, `7`, `3`, `0`, `1`, `*`, and `2`.

### Step 3: Calculate the Sum of the Known Digits
Let's calculate the sum of the known digits:

- \( 6 + 7 + 3 + 0 + 1 + 2 = 19 \)

### Step 4: Include the Unknown Digit
Let the unknown digit (represented by `*`) be `x`. Now the sum of the digits becomes:

- \( 19 + x \)

### Step 5: Set Up the Condition for Divisibility by 9
We need \( 19 + x \) to be divisible by `9`. This means we can express this condition mathematically:

- \( (19 + x) \mod 9 = 0 \)

### Step 6: Find the Remainder of 19 When Divided by 9
Calculating \( 19 \mod 9 \):

- \( 19 \div 9 = 2 \) remainder \( 1 \)

So, \( 19 \equiv 1 \mod 9 \).

### Step 7: Determine the Value of x
To make \( 19 + x \equiv 0 \mod 9 \), we need:

- \( 1 + x \equiv 0 \mod 9 \)

This simplifies to:

- \( x \equiv -1 \mod 9 \)
- \( x \equiv 8 \mod 9 \)

### Step 8: Conclusion
The least non-negative integer that satisfies this condition is \( x = 8 \).

Thus, the least number that should replace `**` is **8**.

### Final Answer
The least number that should be replaced for `**` is **8**.

---