Let N=21224-1, α=2153+277+1 and β=2408-2204+1. Then which of the following statement is correct?

Question

Let N=21224-1, α=2153+277+1 and β=2408-2204+1. Then which of the following statement is correct? (A) α divides N but β does not (B) β divides N but α does not (C) α and β both divides N (D) neither α nor β divides N

Tardigrade Admin · Accepted Answer

Let x=2(153/2) ∴ α=x2+√2 x+1 Let y=2204 ∴ β=y2-y+1 ⇒ N=x16-1 ⇒ N=(x4-1)(x4+1)(x8+1) ⇒ N=(x4-1)(x2+√2 x+1)(x2-√2 x+1)(x8+1) ⇒ N=y6-1=(y3-1)(y3+1)=(y3-1)(y+1)(y2-y+1)

Q. Let $N = 2^{1224} - 1, α = 2^{153} + 2^{77} + 1$ and $β = 2^{408} - 2^{204} + 1$ . Then which of the following statement is correct?

$α$ divides $N$ but $β$ does not

$β$ divides $N$ but $α$ does not

$α$ and $β$ both divides $N$

neither $α$ nor $β$ divides $N$

Q. Let N=21224−1,α=2153+277+1 and β=2408−2204+1. Then which of the following statement is correct?

α divides N but β does not

β divides N but α does not

α and β both divides N

neither α nor β divides N

Let x=22153​∴α=x2+2​x+1 Let y=2204∴β=y2−y+1 ⇒N=x16−1 ⇒N=(x4−1)(x4+1)(x8+1) ⇒N=(x4−1)(x2+2​x+1)(x2−2​x+1)(x8+1) ⇒N=y6−1=(y3−1)(y3+1)=(y3−1)(y+1)(y2−y+1)

Q. Let $N = 2^{1224} - 1, α = 2^{153} + 2^{77} + 1$ and $β = 2^{408} - 2^{204} + 1$ . Then which of the following statement is correct?

$α$ divides $N$ but $β$ does not

$β$ divides $N$ but $α$ does not

$α$ and $β$ both divides $N$

neither $α$ nor $β$ divides $N$