The last three-digits of the multiplication `12345 ×x 54321`

Options

A. 865
B. 745
C. 845
D. 945

Answer

To find the last three digits of the multiplication \( 12345 \times 54321 \), we can follow these steps:

### Step 1: Write down the numbers
We have:
- First number: \( 12345 \)
- Second number: \( 54321 \)

### Step 2: Multiply the numbers
To find the last three digits of the product, we can perform the multiplication directly but focus only on the last three digits. We can do this by performing the multiplication step by step.

### Step 3: Break down the multiplication
We can multiply the numbers as follows:
1. Multiply \( 12345 \) by \( 1 \) (the last digit of \( 54321 \)):
\[ 12345 \times 1 = 12345 \]
Last three digits: \( 345 \)

2. Multiply \( 12345 \) by \( 2 \) (the second last digit of \( 54321 \)):
\[ 12345 \times 2 = 24690 \]
Last three digits: \( 690 \)

3. Multiply \( 12345 \) by \( 3 \) (the third last digit of \( 54321 \)):
\[ 12345 \times 3 = 37035 \]
Last three digits: \( 035 \)

4. Multiply \( 12345 \) by \( 4 \) (the fourth last digit of \( 54321 \)):
\[ 12345 \times 4 = 49380 \]
Last three digits: \( 380 \)

5. Multiply \( 12345 \) by \( 5 \) (the fifth last digit of \( 54321 \)):
\[ 12345 \times 5 = 61725 \]
Last three digits: \( 725 \)

### Step 4: Combine the results
Now we need to add the last three digits from each multiplication, but we will only keep track of the last three digits at each step:

- From \( 1 \): \( 345 \)
- From \( 2 \): \( 690 \)
- From \( 3 \): \( 035 \)
- From \( 4 \): \( 380 \)
- From \( 5 \): \( 725 \)

Adding these together:
\[ 345 + 690 + 35 + 380 + 725 = 2175 \]

### Step 5: Find the last three digits
The last three digits of \( 2175 \) are \( 175 \).

### Final Answer
Thus, the last three digits of the multiplication \( 12345 \times 54321 \) are \( 175 \).