[tex]U(99^{51}) = U(9^{51}) = U(9^{2\cdot 25+1}) = U(9^{2\cdot 25}\cdot 9) = \\ \\ = U(81^{25}\cdot 9) =U(1\cdot 9) = \boxed{9}\\ \\ \\U(11^{53}+15^{53}+17^{53}) = U(1+5+7^{53}) = \\ \\ = U(6+7^{2\cdot 26+1})=U(6+49^{26}\cdot 7) =U(6+9^{26}\cdot 7)\\ \\ =U(6+81^{13}\cdot 7) =U(6+1\cdot 7) = U(13) = \boxed{3}\\ \\ \\ U(313^{100}) = U(3^{100}) = U(9^{50}) = U(81^{25}) = \boxed{1}[/tex]
