Find the unit digit of 111!. (factorial 111) .

Answer

To find the unit digit of \( 111! \) (111 factorial), we can follow these steps:

### Step 1: Understanding Factorial
The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

### Step 2: Express \( 111! \)
We can express \( 111! \) as:
\[ 111! = 111 \times 110 \times 109 \times 108 \times \ldots \times 1 \]

### Step 3: Identify Multiples of 10
In the product \( 111! \), we need to check for the presence of multiples of 10. A multiple of 10 is formed by multiplying 2 and 5.

### Step 4: Count Factors of 5
To determine how many times 10 can be formed, we need to count the number of factors of 5 in \( 111! \):
- The multiples of 5 up to 111 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110.
- There are 22 multiples of 5.

### Step 5: Count Factors of 2
Next, we count the factors of 2:
- The multiples of 2 up to 111 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110.
- There are many more factors of 2 than factors of 5.

### Step 6: Conclusion
Since there are more factors of 2 than factors of 5, we can conclude that there are enough pairs of 2 and 5 to form multiples of 10 in \( 111! \). Therefore, \( 111! \) will have at least one factor of 10, which means the unit digit of \( 111! \) is 0.

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

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