[Toán 8] Tìm số dư

I

icy_tears

Cách này dài, hi vọng đúng ^^

Ta có:
$5^{1991} + 5^{1999} + 5^{2007}$

$= 5^3(5^{1988} + 5^{1996} + 5^{2004})$

$= 125 . [(5^4)^{497} + (5^4)^{499} + (5^4)^{501}]$

$\equiv 125 . (158^{497} + 158^499 + 158^501)$ (mod 467)

$\equiv 125 . 158^{497} . (1 + 158^2 + 158^4)$ (mod 467)

$\equiv 125 . (158^2)^{248} . 158 . 284)$ (mod 467)

$\equiv 330 . 213^{248}$ (mod 467)

$\equiv 330 . 70^{124}$ (mod 467)

$\equiv 330 . (70^4)^{31}$ (mod 467)

$\equiv 330 . 129^{31}$ (mod 467)

$\equiv 330 . 129 . 129^{30}$ (mod 467)

$\equiv 73 . (129^3)^{10}$ (mod 467)

$\equiv 73 . 357^{10}$ (mod 467)

$\equiv 73 . 357 . 357^{9}$ (mod 467)

$\equiv 376 . 417^3$ (mod 467)

$\equiv 376 . 156$ (mod 467)

$\equiv 215$ (mod 467)
 
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