Question

Sa se afle restul impartirii lui $439 ^{72} ~la~13.$

Answer 1

[tex]439^{72}=(13\cdot 33+10)^{72}=13k+10^{72}\\ Din\ teo\ lui\ Fermat:(10^{13}-10)\vdots13\rightarrow10^{13}=13t+10\\ 10^{72}=(10^{13})^5\cdot 10^{7}=(13t+10)^{5}\cdot10^7=(13u+10^5)\cdot10^7=\\ =13z+10^{12}=13z+100^6=13z+(13\cdot7+9)^6=\\ =13z+13w+9^6=13q+81^3=13q+(13\cdot6+3)^3=\\ =13q+13s+27=13h+13\cdot2+1=13x+1\\ 439^{72}=13k+13x+1=13m+1\\ 439^{72}:13=m\ rest\ 1[/tex]