Let N_1 = 2^{55} + 1 and N_2 = 165. Then: | vedclass

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(D) Given $N_2 = 165 = 3 	imes 5 	imes 11$ and $N_1 = 2^{55} + 1$.We know that if $n$ is an odd integer,then $x^n + y^n$ is divisible by $x + y$.Sin

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the $HCF$ of $N_1$ and $N_2$ is $33$

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(D) Given $N_2 = 165 = 3 	imes 5 	imes 11$ and $N_1 = 2^{55} + 1$.We know that if $n$ is an odd integer,then $x^n + y^n$ is divisible by $x + y$.Since $55$ is odd,$N_1 = 2^{55} + 1^{55}$ is divisible by $2 + 1 = 3$.Also,$N_1 = (2^5)^{11} + 1^{11} = 32^{11} + 1^{11}$,which is divisible by $32 + 1 = 33$.Since $33 = 3 	imes 11$,$N_1$ is divisible by both $3$ and $11$.$N_2 = 165 = 15 	imes 11 = 5 	imes 33$.Since $N_1$ is a multiple of $33$ and $N_2$ is a multiple of $33$,and $N_1$ is not divisible by $5$ (as $2^{55} + 1 \equiv (2^2)^{27} 	imes 2 + 1 \equiv (-1)^{27} 	imes 2 + 1 \equiv -2 + 1 \equiv -1 \equiv 4 \pmod{5}$),the $HCF$ of $N_1$ and $N_2$ is $33$.

Let $N_1 = 2^{55} + 1$ and $N_2 = 165$. Then:

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