Find out the unit digit in the -
`(1xx2xx3xx4xx5xx6xx7xx8xx9xx10)/100`

Options

A. 7
B. 9
C. 8
D. None of these

Answer

To find the unit digit of the expression \((1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10)/100\), we can follow these steps:

### Step 1: Simplify the expression
First, we note that the expression is divided by 100. Since 100 can be factored as \(10 \times 10\), we can see that the factors of 10 in the numerator will cancel out with the denominator.

### Step 2: Identify the factors of 10
The factors of 10 in the numerator come from the numbers 2 and 5. In the product \(1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10\), we have:
- One 2 from \(2\)
- One 5 from \(5\)
- One additional 2 from \(10\) (which is \(2 \times 5\))

Thus, we have a total of two factors of 10 in the numerator, which means we can cancel out \(10\) from the numerator twice.

### Step 3: Remaining factors after cancellation
After canceling \(10\) twice, we are left with the product of the remaining numbers:
\[ 1 \times 2 \times 3 \times 4 \times 6 \times 7 \times 8 \times 9 \]

### Step 4: Calculate the unit digit of the remaining product
Now we need to find the unit digit of the product \(1 \times 2 \times 3 \times 4 \times 6 \times 7 \times 8 \times 9\).

- The unit digits of the numbers are:
- \(1\) (unit digit is \(1\))
- \(2\) (unit digit is \(2\))
- \(3\) (unit digit is \(3\))
- \(4\) (unit digit is \(4\))
- \(6\) (unit digit is \(6\))
- \(7\) (unit digit is \(7\))
- \(8\) (unit digit is \(8\))
- \(9\) (unit digit is \(9\))

### Step 5: Multiply the unit digits
Now we will multiply these unit digits together and keep track of the unit digit at each step:
1. \(1 \times 2 = 2\) (unit digit is \(2\))
2. \(2 \times 3 = 6\) (unit digit is \(6\))
3. \(6 \times 4 = 24\) (unit digit is \(4\))
4. \(4 \times 6 = 24\) (unit digit is \(4\))
5. \(4 \times 7 = 28\) (unit digit is \(8\))
6. \(8 \times 8 = 64\) (unit digit is \(4\))
7. \(4 \times 9 = 36\) (unit digit is \(6\))

### Step 6: Conclusion
The unit digit of the entire expression \((1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10)/100\) is \(6\).