Question

which one of the following listed numbers is not a divisor of the number N= (2³⁰-1), is equal to

(a) 2⁵-1 (b) 2⁵+1 (c) 2⁶-1 (d) 2¹⁰+1

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{number}\mathrm{is} \\ \mathrm{N}=2^{30}-1 \\ \Rightarrow \mathrm{N}={\left(2^{10}\right)}^3-{\left(1\right)}^3 \\ \Rightarrow \mathrm{N}=\left(2^{10}-1\right)\left\{{\left(2^{10}\right)}^2+{\left(1\right)}^2+2^{10}.1\right\}\mathbf{}\mathbf{}\left[\mathbf{using}\mathit{}{\mathit{a}}^{\mathbf{3}}\mathbf{-}{\mathit{b}}^{\mathbf{3}}\mathbf{}\mathbf{=}\mathbf{}\left(\mathit{a}\mathbf{-}\mathit{b}\right)\mathbf{}\left({\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathit{b}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathit{}\mathit{a}\mathit{b}\right)\right] \\ \Rightarrow \mathrm{N}=\left(2^{10}-1\right)\left(2^{20}+1+2^{10}\right) \\ \Rightarrow \mathrm{N}=\left\{{\left(2^5\right)}^2-{\left(1\right)}^2\right\}\left(2^{20}+1+2^{10}\right) \\ \Rightarrow \mathrm{N}=\left(2^5-1\right)\left(2^5+1\right)\left(2^{20}+1+2^{10}\right)\mathbf{}\mathbf{}\left[\mathbf{using}\mathbf{}{\mathit{a}}^{\mathbf{2}}\mathbf{-}{\mathit{b}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\left(\mathit{a}\mathbf{-}\mathit{b}\right)\mathbf{}\left(\mathit{a}\mathbf{+}\mathit{b}\right)\right] \\ \mathrm{Hence}{2}^5-1,2^5+1\mathrm{and}{2}^{10}-1\mathrm{are}\mathrm{the}\mathrm{factors}\mathrm{of}\mathrm{given}\mathrm{number}\mathrm{N}. \\ \mathrm{But}{2}^6-1\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{factor}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{number} \\ \mathrm{Hence},2^6-1\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{divisor}\mathrm{of}\mathrm{given}\mathrm{number}\mathrm{N}. \end{gathered}$$