Question

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?

• A. $\alpha$ divides $N$ but $\beta$ does not
• B. $\beta$ divides $N$ but $\alpha$ does not
• C. $\alpha$ and $\beta$ both divides $N$
• D. neither $\alpha$ nor $\beta$ divides $N$

Answer 1

Answer: (C)

Let $x=2^{\frac{153}{2}} \therefore \alpha=x^{2}+\sqrt{2} x+1$
Let $y=2^{204} \therefore \beta=y^{2}-y+1$
$\Rightarrow N=x^{16}-1 $
$\Rightarrow N=\left(x^{4}-1\right)\left(x^{4}+1\right)\left(x^{8}+1\right)$
$\Rightarrow N=\left(x^{4}-1\right)\left(x^{2}+\sqrt{2} x+1\right)\left(x^{2}-\sqrt{2} x+1\right)\left(x^{8}+1\right)$
$\Rightarrow N=y^{6}-1=\left(y^{3}-1\right)\left(y^{3}+1\right)=\left(y^{3}-1\right)(y+1)\left(y^{2}-y+1\right)$