Find unit digit in `(11)^1. (12)^2. (13)^3. (14)^4. (15)^5. (16)^6`.

Options

A. 3
B. 0
C. 2
D. 5

Answer

To find the unit digit of the expression \( (11)^1 \cdot (12)^2 \cdot (13)^3 \cdot (14)^4 \cdot (15)^5 \cdot (16)^6 \), we can follow these steps:

### Step 1: Identify the unit digits of the bases
The unit digits of the bases are:
- \( 11 \) has a unit digit of \( 1 \)
- \( 12 \) has a unit digit of \( 2 \)
- \( 13 \) has a unit digit of \( 3 \)
- \( 14 \) has a unit digit of \( 4 \)
- \( 15 \) has a unit digit of \( 5 \)
- \( 16 \) has a unit digit of \( 6 \)

### Step 2: Calculate the unit digit of each term
Now we will calculate the unit digit of each term raised to its respective power:

1. **For \( 11^1 \)**:
- Unit digit = \( 1^1 = 1 \)

2. **For \( 12^2 \)**:
- Unit digit = \( 2^2 = 4 \)

3. **For \( 13^3 \)**:
- Unit digit = \( 3^3 = 27 \) → Unit digit = \( 7 \)

4. **For \( 14^4 \)**:
- \( 4^4 = 256 \) → Unit digit = \( 6 \)

5. **For \( 15^5 \)**:
- Unit digit = \( 5^5 = 3125 \) → Unit digit = \( 5 \)

6. **For \( 16^6 \)**:
- Unit digit = \( 6^6 = 46656 \) → Unit digit = \( 6 \)

### Step 3: Combine the unit digits
Now we will multiply the unit digits we found:
- From \( 11^1 \): \( 1 \)
- From \( 12^2 \): \( 4 \)
- From \( 13^3 \): \( 7 \)
- From \( 14^4 \): \( 6 \)
- From \( 15^5 \): \( 5 \)
- From \( 16^6 \): \( 6 \)

So we need to find the unit digit of:
\[ 1 \cdot 4 \cdot 7 \cdot 6 \cdot 5 \cdot 6 \]

### Step 4: Calculate the unit digit step by step
1. \( 1 \cdot 4 = 4 \) (unit digit = 4)
2. \( 4 \cdot 7 = 28 \) (unit digit = 8)
3. \( 8 \cdot 6 = 48 \) (unit digit = 8)
4. \( 8 \cdot 5 = 40 \) (unit digit = 0)
5. \( 0 \cdot 6 = 0 \) (unit digit = 0)

### Final Result
The unit digit of the entire expression \( (11)^1 \cdot (12)^2 \cdot (13)^3 \cdot (14)^4 \cdot (15)^5 \cdot (16)^6 \) is \( 0 \).

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