What is the unit's digit of `125^(125)+216^(216)` ?

Options

A. 5
B. 1
C. 6
D. 7

Answer

To find the unit's digit of the expression \( 125^{125} + 216^{216} \), we can break it down into two parts: finding the unit's digit of \( 125^{125} \) and \( 216^{216} \).

### Step 1: Find the unit's digit of \( 125^{125} \)

The unit's digit of any number raised to a power depends only on the unit's digit of the base. The unit's digit of \( 125 \) is \( 5 \).

- The powers of \( 5 \) always end with \( 5 \):
- \( 5^1 = 5 \)
- \( 5^2 = 25 \)
- \( 5^3 = 125 \)
- \( 5^4 = 625 \)

Thus, the unit's digit of \( 125^{125} \) is \( 5 \).

### Step 2: Find the unit's digit of \( 216^{216} \)

Next, we look at \( 216 \). The unit's digit of \( 216 \) is \( 6 \).

- The powers of \( 6 \) always end with \( 6 \):
- \( 6^1 = 6 \)
- \( 6^2 = 36 \)
- \( 6^3 = 216 \)
- \( 6^4 = 1296 \)

Thus, the unit's digit of \( 216^{216} \) is \( 6 \).

### Step 3: Add the unit's digits

Now we can add the unit's digits we found:
- The unit's digit of \( 125^{125} \) is \( 5 \).
- The unit's digit of \( 216^{216} \) is \( 6 \).

Adding these together:
\[ 5 + 6 = 11 \]

The unit's digit of \( 11 \) is \( 1 \).

### Final Answer

Therefore, the unit's digit of \( 125^{125} + 216^{216} \) is \( 1 \).

---