Find the difference between the largest 7 - digit number and the smallest 7 - digit number formed by using the digits `4, 0, 3, 9` and 7 ( using each digit at least once ).

Options

A. `6993951`
B. `6949530`
C. `6963951`
D. `6996951`

Answer

To find the difference between the largest 7-digit number and the smallest 7-digit number formed by using the digits 4, 0, 3, 9, and 7 (using each digit at least once), we will follow these steps:

### Step 1: Identify the digits
The digits we have are: 4, 0, 3, 9, and 7.

### Step 2: Form the largest 7-digit number
To form the largest 7-digit number, we should use the largest digits in descending order. Since we need to use each digit at least once, we can repeat the largest digit (which is 9) to fill the remaining places.

- The largest digit is 9. We can use it three times.
- The next largest digits are 7, 4, 3, and 0.

So, the largest 7-digit number will be:
**9997430**

### Step 3: Form the smallest 7-digit number
To form the smallest 7-digit number, we should use the smallest digits in ascending order. However, we cannot start with 0, as that would make it a 6-digit number. Therefore, we will start with the smallest non-zero digit.

- The smallest non-zero digit is 3. We will start with 3.
- Next, we can use 0, then 4, 7, and 9. To fill the remaining two places, we will repeat the smallest digit, which is 0.

So, the smallest 7-digit number will be:
**3000049**

### Step 4: Calculate the difference
Now, we will calculate the difference between the largest and smallest 7-digit numbers.

\[ \text{Largest number} = 9997430 \]
\[ \text{Smallest number} = 3000049 \]

Now, we perform the subtraction:

\[ 9997430 - 3000049 \]

### Step 5: Perform the subtraction
Let's perform the subtraction step by step:

```
9997430
- 3000049
-----------
```

Starting from the rightmost digit:

1. **0 - 9**: We need to borrow. So, we take 1 from the next column (which is 3), making it 2, and we add 10 to 0, making it 10. Thus, 10 - 9 = 1.
2. **2 - 4**: We need to borrow again. So, we take 1 from the next column (which is 4), making it 3, and we add 10 to 2, making it 12. Thus, 12 - 4 = 8.
3. **3 - 0**: This is straightforward. 3 - 0 = 3.
4. **7 - 0**: This is straightforward. 7 - 0 = 7.
5. **4 - 0**: This is straightforward. 4 - 0 = 4.
6. **9 - 0**: This is straightforward. 9 - 0 = 9.
7. **9 - 3**: This is straightforward. 9 - 3 = 6.

Putting it all together, we get:

\[ 9997430 - 3000049 = 6997381 \]

### Final Answer
The difference between the largest 7-digit number and the smallest 7-digit number is **6997381**.

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