The digit in unit's place of the product `81 xx 82 xx 83 xx`... `xx 89` is

Options

A. 0
B. 2
C. 6
D. 8

Answer

To find the digit in the unit's place of the product \( 81 \times 82 \times 83 \times \ldots \times 89 \), we can follow these steps:

### Step 1: Identify the unit digits of the numbers
First, we need to identify the unit digits of each number from 81 to 89:
- The unit digit of 81 is **1**
- The unit digit of 82 is **2**
- The unit digit of 83 is **3**
- The unit digit of 84 is **4**
- The unit digit of 85 is **5**
- The unit digit of 86 is **6**
- The unit digit of 87 is **7**
- The unit digit of 88 is **8**
- The unit digit of 89 is **9**

### Step 2: Write down the unit digits
The unit digits we have are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9 \]

### Step 3: Calculate the product of the unit digits
Now, we will multiply these unit digits together:
\[ 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \]

### Step 4: Identify the presence of 2 and 5
In this product, we can see that there is a **2** (from 2, 4, 6, 8) and a **5** (from 5). When we multiply 2 and 5 together, we get:
\[ 2 \times 5 = 10 \]

### Step 5: Determine the unit digit of the product
Since \( 10 \) ends with a **0**, any product that includes a factor of 10 will also end with **0**. Thus, the unit digit of the entire product \( 81 \times 82 \times 83 \times \ldots \times 89 \) is:
\[ \text{Unit digit} = 0 \]

### Final Answer
The digit in the unit's place of the product \( 81 \times 82 \times 83 \times \ldots \times 89 \) is **0**.

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