Question Number 201418 by cortano12 last updated on 06/Dec/23
$$\:\:\:\:\:\:\mathrm{2025}^{\mathrm{2025}} \:=\:\mathrm{x}\:\left(\mathrm{mod}\:\mathrm{17}\:\right) \\ $$
Answered by mr W last updated on 06/Dec/23
$$\mathrm{2025}^{\mathrm{2025}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$=\left(\mathrm{119}×\mathrm{17}+\mathrm{2}\right)^{\mathrm{2025}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$\equiv\mathrm{2}^{\mathrm{2025}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$=\mathrm{2}×\mathrm{2}^{\mathrm{4}×\mathrm{506}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$=\mathrm{2}×\left(\mathrm{17}−\mathrm{1}\right)^{\mathrm{506}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$\equiv\mathrm{2}×\left(−\mathrm{1}\right)^{\mathrm{506}} \:\left({mod}\:\mathrm{17}\right) \\ $$$$\equiv\mathrm{2}\:\left({mod}\:\mathrm{17}\right) \\ $$
Answered by BaliramKumar last updated on 06/Dec/23
![((2025^(2025) )/(17)) [ φ(17)= 16 ] (((17×119+2)^(16×126+9) )/(17)) (2^9 /(17)) = (((2^4 )^2 ×2^1 )/(17)) = (((16)^2 ×2)/(17)) (((17−1)^2 ×2)/(17)) = (((−1)^2 ×2)/(17)) ((1×2)/(17)) = (2/(17)) = 2](https://www.tinkutara.com/question/Q201423.png)
$$\frac{\mathrm{2025}^{\mathrm{2025}} }{\mathrm{17}}\:\:\:\:\:\:\:\:\:\:\:\:\left[\:\phi\left(\mathrm{17}\right)=\:\mathrm{16}\:\right] \\ $$$$\:\frac{\left(\mathrm{17}×\mathrm{119}+\mathrm{2}\right)^{\mathrm{16}×\mathrm{126}+\mathrm{9}} }{\mathrm{17}}\:\: \\ $$$$\frac{\mathrm{2}^{\mathrm{9}} }{\mathrm{17}}\:=\:\frac{\left(\mathrm{2}^{\mathrm{4}} \right)^{\mathrm{2}} ×\mathrm{2}^{\mathrm{1}} }{\mathrm{17}}\:=\:\:\frac{\left(\mathrm{16}\right)^{\mathrm{2}} ×\mathrm{2}}{\mathrm{17}} \\ $$$$\:\:\frac{\left(\mathrm{17}−\mathrm{1}\right)^{\mathrm{2}} ×\mathrm{2}}{\mathrm{17}}\:=\:\frac{\left(−\mathrm{1}\right)^{\mathrm{2}} ×\mathrm{2}}{\mathrm{17}}\: \\ $$$$\frac{\mathrm{1}×\mathrm{2}}{\mathrm{17}}\:=\:\frac{\mathrm{2}}{\mathrm{17}}\:=\:\mathrm{2} \\ $$$$ \\ $$