a) Ta thấy: [tex]2^x+5^y\equiv (-1)^x+(-1)^y(mod3);19^z\equiv 1^z\equiv 1(mod3)\Rightarrow (-1)^x+(-1)^y\equiv 1\equiv -2(mod3)\Rightarrow x,y[/tex] lẻ [tex]\Rightarrow y\geq 1\Rightarrow 19^z-2^x=5^y\vdots 5\Rightarrow 19^z\equiv 2^x(mod5)\Rightarrow (-1)^z\equiv 2^x(mod5)[/tex]
Vì [tex]2^x=2^{2k+1}=4^k.2\equiv (-1)^k.2\equiv \pm 2\not\equiv (-1)^z(mod5)\Rightarrow[/tex] Phương trình vô nghiệm.
b) Ta có: [tex]2^y=3^x-1\equiv (-1)^x-1(mod4)[/tex]
Xét y = 0 [tex]\Rightarrow 3^x=2[/tex](loại)
Xét y = 1 [tex]\Rightarrow 3^x=3\Rightarrow x=1[/tex]
Với [tex]y\geq 2\Rightarrow 2^y\vdots 4\Rightarrow (-1)^x-1=0\Rightarrow x=2k[/tex]
[tex]\Rightarrow 2^y=3^{2k}-1=(3^k-1)(3^k+1)\Rightarrow \left\{\begin{matrix} 3^k-1=2^m\\ 3^k+1=2^n \end{matrix}\right.\Rightarrow 2=2^n-2^m\Rightarrow 2^m(2^{n-m}-1)=2\Rightarrow \left\{\begin{matrix} 2^m=2\\ 2^{n-m}-1=1 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} n=2\\ m=1 \end{matrix}\right.\Rightarrow y=3[/tex]
Vậy [tex](x,y)=(1,1);(2,3)[/tex]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
