The units' digit in the expression `(11^(1)+12^(2)+13^(3)+14^(4)+15^(5)+16^(6))` is

Options

A. 1
B. 7
C. 9
D. 0

Answer

To find the units digit of the expression \(11^{1} + 12^{2} + 13^{3} + 14^{4} + 15^{5} + 16^{6}\), we will calculate the units digit of each term separately and then sum them up.

### Step-by-step Solution:

1. **Calculate the units digit of \(11^{1}\)**:
- The units digit of \(11\) is \(1\).
- Therefore, \(11^{1}\) has a units digit of \(1\).

2. **Calculate the units digit of \(12^{2}\)**:
- The units digit of \(12\) is \(2\).
- Now, calculate \(2^{2} = 4\).
- Therefore, the units digit of \(12^{2}\) is \(4\).

3. **Calculate the units digit of \(13^{3}\)**:
- The units digit of \(13\) is \(3\).
- Now, calculate \(3^{3} = 27\).
- Therefore, the units digit of \(13^{3}\) is \(7\).

4. **Calculate the units digit of \(14^{4}\)**:
- The units digit of \(14\) is \(4\).
- The units digits of powers of \(4\) cycle every two terms: \(4^1 = 4\) and \(4^2 = 6\).
- Since \(4\) is even, the units digit of \(14^{4}\) is \(6\).

5. **Calculate the units digit of \(15^{5}\)**:
- The units digit of \(15\) is \(5\).
- The units digit of any power of \(5\) is always \(5\) (for odd powers).
- Therefore, the units digit of \(15^{5}\) is \(5\).

6. **Calculate the units digit of \(16^{6}\)**:
- The units digit of \(16\) is \(6\).
- The units digit of any power of \(6\) is always \(6\).
- Therefore, the units digit of \(16^{6}\) is \(6\).

7. **Sum the units digits**:
- Now, we sum the units digits we calculated:
\[ 1 + 4 + 7 + 6 + 5 + 6 = 29 \]

8. **Find the units digit of the sum**:
- The units digit of \(29\) is \(9\).

### Final Answer:
The units digit in the expression \(11^{1} + 12^{2} + 13^{3} + 14^{4} + 15^{5} + 16^{6}\) is \(9\).

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