What is the last digit of `3^(4n) +1`, where n is a natural number ?
(a)2
(b)7
(c)8
(d)None of these

Options

A. 2
B. 7
C. 8
D. None of these

Answer

To find the last digit of \(3^{4n} + 1\) where \(n\) is a natural number, we can follow these steps:

### Step 1: Understand the expression
We need to determine the last digit of the expression \(3^{4n} + 1\). The last digit of a number can be found by evaluating it modulo 10.

### Step 2: Find the last digit of \(3^{4n}\)
To find the last digit of \(3^{4n}\), we first need to observe the pattern in the last digits of powers of 3:
- \(3^1 = 3\) (last digit is 3)
- \(3^2 = 9\) (last digit is 9)
- \(3^3 = 27\) (last digit is 7)
- \(3^4 = 81\) (last digit is 1)
- \(3^5 = 243\) (last digit is 3)

From this, we can see that the last digits of powers of 3 repeat every 4 terms: \(3, 9, 7, 1\).

### Step 3: Determine \(4n \mod 4\)
Since the last digits repeat every 4 powers, we can find \(4n \mod 4\):
- \(4n \mod 4 = 0\) for any natural number \(n\).

### Step 4: Find the last digit corresponding to \(4n \equiv 0 \mod 4\)
From our earlier observation, when \(n\) is such that \(4n \equiv 0 \mod 4\), the last digit of \(3^{4n}\) corresponds to the last digit of \(3^4\), which is 1.

### Step 5: Calculate \(3^{4n} + 1\)
Now we can calculate:
\[ 3^{4n} + 1 \equiv 1 + 1 \equiv 2 \mod 10 \]

### Conclusion
Thus, the last digit of \(3^{4n} + 1\) is 2.

### Final Answer
The correct option is (a) 2.

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