What is the unit's digit of `29^(136)` ?

Options

A. 1
B. 3
C. 7
D. 9

Answer

To find the unit's digit of \(29^{136}\), we can simplify the problem by focusing on the unit's digit of the base, which is \(9\). Therefore, we need to find the unit's digit of \(9^{136}\).

### Step 1: Identify the pattern in the unit's digits of powers of 9
Let's calculate the unit's digits of the first few powers of \(9\):
- \(9^1 = 9\) (unit's digit is **9**)
- \(9^2 = 81\) (unit's digit is **1**)
- \(9^3 = 729\) (unit's digit is **9**)
- \(9^4 = 6561\) (unit's digit is **1**)

From this, we can see that the unit's digits of powers of \(9\) repeat every 2 terms:
- Odd powers of \(9\) have a unit's digit of **9**.
- Even powers of \(9\) have a unit's digit of **1**.

### Step 2: Determine whether the exponent is odd or even
Since \(136\) is an even number, we can conclude that the unit's digit of \(9^{136}\) will be the same as that of \(9^2\).

### Step 3: Find the unit's digit of \(9^2\)
From our earlier calculation:
- The unit's digit of \(9^2\) is **1**.

### Conclusion
Thus, the unit's digit of \(29^{136}\) is **1**.

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