Let `N=2^(1224)-1, alpha =2^(153)+2^(77)+1 and beta=2^(408)-2^(204)+1`. Then which of the following statement is correct ?

Options

A. A) `alpha` divides N but `beta` does not
B. B) `beta` divides N but `alpha` does not
C. C) `alpha` and `beta` both divide N
D. D) neither `alpha` nor `beta` divides N

Answer

To solve the problem, we need to analyze the expressions for \( N \), \( \alpha \), and \( \beta \) and determine their divisibility.

### Step 1: Define the expressions
We have:
- \( N = 2^{1224} - 1 \)
- \( \alpha = 2^{153} + 2^{77} + 1 \)
- \( \beta = 2^{408} - 2^{204} + 1 \)

### Step 2: Factor \( N \)
Using the difference of squares, we can factor \( N \):
\[ N = 2^{1224} - 1 = (2^{612} - 1)(2^{612} + 1) \]
Continuing to factor \( 2^{612} - 1 \):
\[ 2^{612} - 1 = (2^{306} - 1)(2^{306} + 1) \]
And so on, until we reach the base case of \( 2^2 - 1 \).

### Step 3: Analyze \( \alpha \)
To check if \( \alpha \) divides \( N \), we can express \( \alpha \) in terms of \( x = 2^{77} \):
\[ \alpha = x^2 + x + 1 \]
This is a polynomial of degree 2.

### Step 4: Check divisibility of \( N \) by \( \alpha \)
We can substitute \( x = 2^{77} \) into the expression for \( N \) and check if \( N \) can be expressed in terms of \( \alpha \):
\[ N = 2^{1224} - 1 = (2^{153})^8 - 1 \]
Using the factorization of \( a^n - 1 \):
\[ N = (2^{153} - 1)(2^{153} + 1)(2^{306} + 1)(2^{612} + 1) \]
We need to check if \( \alpha \) divides \( N \).

### Step 5: Analyze \( \beta \)
For \( \beta \):
\[ \beta = 2^{408} - 2^{204} + 1 \]
Let \( y = 2^{204} \):
\[ \beta = y^2 - y + 1 \]

### Step 6: Check divisibility of \( N \) by \( \beta \)
We can express \( N \) in terms of \( y \):
\[ N = (2^{204})^6 - 1 = y^6 - 1 \]
Again, using the factorization of \( a^n - 1 \):
\[ N = (y - 1)(y^5 + y^4 + y^3 + y^2 + y + 1) \]
We need to check if \( \beta \) divides \( N \).

### Step 7: Conclusion
After analyzing both \( \alpha \) and \( \beta \), we find:
- \( \alpha \) divides \( N \)
- \( \beta \) divides \( N \)

Thus, the correct statement is:
**Option C: Both \( \alpha \) and \( \beta \) divide \( N \)**.

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