Question

Aflați rădăcina digitala a numărului 2020^2020.

Answer 1

$\displaystyle\bf\\\boxed{\bf TEOREMA~:~ dr(k)=k~mod~9}~.\\\\--------------------\\dr(2020^{2020})=2020^{2020}mod~9.\\2020 \equiv 4(mod~9)~\bigg |^{2020}.\\2020^{2020} \equiv 4^{2020}(mod~9).\\4^2 \equiv7(mod~9).\\4^3 \equiv1(mod~9).\\4^4 \equiv4(mod~9).\\4^5 \equiv7(mod~9).\\observam~ca~restul~se~repeta~din~3~in~3.\\4^{n}\equiv 7(mod~9) \implies n\equiv 2(mod~3).\\4^{n}\equiv 1(mod~9) \implies n\equiv 0(mod~3).\\4^{n}\equiv 4(mod~9) \implies n\equiv 1(mod~3).$

$\displaystyle\bf\\2020\equiv1(mod~3)~\implies asadar~2020^{2020} \equiv 4(mod~9) \implies \boxed{\bf dr(2020^{2020})=4}~.$