If n is any positive interger then expression `3^(4n)-4^(3n)` is exactly divisible?

Options

A. 7
B. 12
C. 17
D. 145

Answer

To determine if the expression \(3^{4n} - 4^{3n}\) is exactly divisible by any of the given options, we can analyze the expression step by step.

### Step 1: Substitute values for \(n\)
Let's start by substituting \(n = 1\) into the expression:

\[ 3^{4 \cdot 1} - 4^{3 \cdot 1} = 3^4 - 4^3 \]

Calculating the powers:

\[ 3^4 = 81 \quad \text{and} \quad 4^3 = 64 \]

So,

\[ 3^4 - 4^3 = 81 - 64 = 17 \]

### Step 2: Substitute \(n = 2\)
Next, we substitute \(n = 2\):

\[ 3^{4 \cdot 2} - 4^{3 \cdot 2} = 3^8 - 4^6 \]

Calculating the powers:

\[ 3^8 = 6561 \quad \text{and} \quad 4^6 = 4096 \]

So,

\[ 3^8 - 4^6 = 6561 - 4096 = 2465 \]

### Step 3: Factor the expression
We can factor the expression \(3^{4n} - 4^{3n}\) using the difference of squares:

\[ 3^{4n} - 4^{3n} = (3^{2n} - 4^{\frac{3n}{2}})(3^{2n} + 4^{\frac{3n}{2}}) \]

### Step 4: Check divisibility by 17
From our calculations, we found that for \(n = 1\), the result is \(17\) and for \(n = 2\), the result is \(2465\). We can check if \(2465\) is divisible by \(17\):

\[ 2465 \div 17 = 145 \]

Since \(145\) is an integer, \(2465\) is also divisible by \(17\).

### Conclusion
Based on our calculations, we can conclude that the expression \(3^{4n} - 4^{3n}\) is exactly divisible by \(17\) for any positive integer \(n\).

### Final Answer
The expression \(3^{4n} - 4^{3n}\) is exactly divisible by **17**.

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