[tex]\it \mathbf{4.}\ \ 49^{12}+7^{25} = (7^2)^{12}+7^{25}=7^{24}+7^{25}=7^{24}\cdot(1+7)=7^{8\cdot3}\cdot8=\\ \\ =(7^8)^3\cdot2^3=(7^8\cdot2)^3\ =\ cub\ \ perfect[/tex]
[tex]\it \mathbf{5.}\ \ x=2^{3n+1}:8^n+5^{2n+3}:25^{n+1}+3^3=2^{3n+1}:2^{3n}+(5^{2n} \cdot5^3):(5^{2n}\cdot5^2)+\\ \\ + 27= 2+5+27=34\\ \\ x:6=34:6=5\ \ rest\ \ 4[/tex]
[tex]\it \mathbf{6.}\ \ 4^n\cdot3^{n+1}+12^{n+1}-6^n\cdot2^{n+1}=4^n\cdot3^n\cdot3+12^n\cdot12-6^n\cdot2^n\cdot2=\\ \\ =12^n\cdot3+12^n\cdot12-12^n\cdot2= 12^n(3+12-2)=12^n\cdot13\ \in\ M_{13}[/tex]
Deci, numărul dat în enunț se împarte exact la 13.
