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Question Number 201418 by cortano12 last updated on 06/Dec/23

$${2025}^{2025}=\mathrm{x}\left(\mathrm{mod}17\right)$$

Answered by mr W last updated on 06/Dec/23

$${2025}^{2025}\left(mod17\right)$$$$={(119\times 17+2)}^{2025}\left(mod17\right)$$$$\equiv 2^{2025}\left(mod17\right)$$$$=2\times 2^{4\times 506}\left(mod17\right)$$$$=2\times {(17-1)}^{506}\left(mod17\right)$$$$\equiv 2\times {(-1)}^{506}\left(mod17\right)$$$$\equiv 2\left(mod17\right)$$

Answered by BaliramKumar last updated on 06/Dec/23

$$\frac{{2025}^{2025}}{17}[\varphi \left(17\right)=16]$$$$\frac{{(17\times 119+2)}^{16\times 126+9}}{17}$$$$\frac{2^9}{17}=\frac{{\left(2^4\right)}^2\times 2^1}{17}=\frac{{\left(16\right)}^2\times 2}{17}$$$$\frac{{(17-1)}^2\times 2}{17}=\frac{{(-1)}^2\times 2}{17}$$$$\frac{1\times 2}{17}=\frac{2}{17}=2$$$$$$

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