`7^(74) - 4^(74)` is divisible by :

Options

A. A)3
B. B)11
C. C)7
D. D)both (A) and (B)

Answer

To determine if \(7^{74} - 4^{74}\) is divisible by certain numbers, we can use the identity for the difference of powers. The identity states that:

\[ a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \ldots + b^{n-1}) \]

This identity tells us that \(a^n - b^n\) is divisible by \(a - b\) and \(a + b\) when \(n\) is even.

### Step 1: Identify \(a\) and \(b\)
Here, we have:
- \(a = 7\)
- \(b = 4\)
- \(n = 74\) (which is even)

### Step 2: Calculate \(a - b\) and \(a + b\)
Now, we calculate:
- \(a - b = 7 - 4 = 3\)
- \(a + b = 7 + 4 = 11\)

### Step 3: Check divisibility
Since \(n\) is even, we can conclude:
- \(7^{74} - 4^{74}\) is divisible by \(3\) (from \(a - b\))
- \(7^{74} - 4^{74}\) is also divisible by \(11\) (from \(a + b\))

### Step 4: Conclusion
Thus, \(7^{74} - 4^{74}\) is divisible by both \(3\) and \(11\). Therefore, the answer is that \(7^{74} - 4^{74}\) is divisible by both \(3\) and \(11\).

### Final Answer
The correct option is \(D\) (both \(A\) and \(B\)).

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