The greatest possible divisor of `3n^(2n + 3) - 24n - 27` for every `n in N`, which necessarily divides is :

Options

A. 64
B. 24
C. 96
D. none of these

Answer

To find the greatest possible divisor of the expression \(3n^{2n + 3} - 24n - 27\) for every \(n \in \mathbb{N}\), we can follow these steps:

### Step 1: Substitute \(n = 1\)
Let's start by substituting \(n = 1\) into the expression:
\[ 3(1)^{2(1) + 3} - 24(1) - 27 \]
Calculating this gives:
\[ 3(1^5) - 24 - 27 = 3 - 24 - 27 \]
\[ = 3 - 51 = -48 \]

### Step 2: Find the absolute value
Since we are looking for divisors, we take the absolute value:
\[ |-48| = 48 \]

### Step 3: Factor 48
Next, we need to find the factors of 48. The prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]
The factors of 48 are:
\[ 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \]

### Step 4: Substitute \(n = 2\)
Now, let's check for \(n = 2\):
\[ 3(2)^{2(2) + 3} - 24(2) - 27 \]
Calculating this gives:
\[ 3(2^7) - 48 - 27 = 3(128) - 48 - 27 \]
\[ = 384 - 48 - 27 = 384 - 75 = 309 \]

### Step 5: Find the greatest common divisor (GCD)
Now we need to find the GCD of the results from \(n = 1\) and \(n = 2\):
- From \(n = 1\), we have 48.
- From \(n = 2\), we have 309.

The factors of 309 are:
\[ 1, 3, 103, 309 \]

### Step 6: Determine the GCD
Now we find the GCD of 48 and 309. The common factors of 48 and 309 are:
- The only common factor is 3.

### Conclusion
The greatest possible divisor of \(3n^{2n + 3} - 24n - 27\) for every \(n \in \mathbb{N}\) is:
\[ \boxed{3} \]