Find out the unit digit in the 111 !

Options

A. 0
B. 1
C. 5
D. 3

Answer

To find the unit digit of \( 111! \) (111 factorial), we need to understand how factorials work and how the unit digit can be determined.

### Step-by-Step Solution:

1. **Understanding Factorial**:
The factorial of a number \( n \) is the product of all positive integers from 1 to \( n \). So, \( 111! = 1 \times 2 \times 3 \times \ldots \times 111 \).

2. **Identifying Factors of 10**:
The unit digit of a number is influenced by the factors of 10 in its product. Each factor of 10 is made up of a factor of 2 and a factor of 5. Therefore, we need to count how many pairs of 2s and 5s are present in the factorial.

3. **Counting Factors of 5**:
To find the number of factors of 5 in \( 111! \), we use the formula:
\[ \text{Number of factors of 5} = \left\lfloor \frac{111}{5} \right\rfloor + \left\lfloor \frac{111}{25} \right\rfloor \]
- \( \left\lfloor \frac{111}{5} \right\rfloor = 22 \)
- \( \left\lfloor \frac{111}{25} \right\rfloor = 4 \)
- Total factors of 5 = \( 22 + 4 = 26 \)

4. **Counting Factors of 2**:
Similarly, we count the number of factors of 2:
\[ \text{Number of factors of 2} = \left\lfloor \frac{111}{2} \right\rfloor + \left\lfloor \frac{111}{4} \right\rfloor + \left\lfloor \frac{111}{8} \right\rfloor + \left\lfloor \frac{111}{16} \right\rfloor + \left\lfloor \frac{111}{32} \right\rfloor + \left\lfloor \frac{111}{64} \right\rfloor \]
- \( \left\lfloor \frac{111}{2} \right\rfloor = 55 \)
- \( \left\lfloor \frac{111}{4} \right\rfloor = 27 \)
- \( \left\lfloor \frac{111}{8} \right\rfloor = 13 \)
- \( \left\lfloor \frac{111}{16} \right\rfloor = 6 \)
- \( \left\lfloor \frac{111}{32} \right\rfloor = 3 \)
- \( \left\lfloor \frac{111}{64} \right\rfloor = 1 \)
- Total factors of 2 = \( 55 + 27 + 13 + 6 + 3 + 1 = 105 \)

5. **Determining the Number of Pairs**:
Since we have 26 factors of 5 and 105 factors of 2, the number of pairs of (2, 5) that can form 10 is limited by the smaller count, which is 26. Therefore, there are 26 factors of 10 in \( 111! \).

6. **Conclusion**:
Since \( 111! \) has at least one factor of 10, the unit digit of \( 111! \) is 0.

### Final Answer:
The unit digit of \( 111! \) is **0**.