5.Tìm số dư khi chia 2011109 +201267 + 6739543 cho 57
Ta có:
+) $2011^{109}\equiv 16^{109}$ (mod 57)
$\equiv 16^{100}.16^{9}$ (mod 57)
$\equiv (16^{2})^{50}.(16^{3})^{3}$ (mod 57)
$\equiv 28^{50}.49^{3}$ (mod 57)
$\equiv (28^{2})^{25}.1$ (mod 57)
$\equiv 43^{25}.1$ (mod 57)
$\equiv 28^{5}.1$ (mod 57)
$\equiv 16$ (mod 57)
+)$2012^{67}\equiv 17^{67}$ (mod 57)
$\equiv 17^{60}.17^{7}$ (mod 57)
$\equiv 16^{5}.5$ (mod 57)
$\equiv 4^{3}.5$ (mod 57)
$\equiv 35$ (mod 57)
+)$6789123456789= 678912345.10^{4}+6789$
$\equiv 51.10^{4}+6$ (mod 57)
$\equiv 51.25+6$ (mod 57)
$\equiv 27$ (mod 57)
=>số dư khi chia 2011109 +201267 + 6739543 cho 57 là 21.