335. Find the last two digits of the number $3^{5^{17}}$.
335. Answer. 43.

Hint. The last digit of the number $a^{n}(n>4)$ is the same as the last digit of the number $a^{r}$, where $r>0$ is the remainder of the division of $n$ by 4; if $r=0$, then the last digit of the number $a^{n}$ coincides with the last digit of the number $a^{4}$ (see № 89). If $(a, 10)=1$, then the last two digits of the number $a^{n}(n>40)$ are the same as the last two digits of the number $a^{r}$, where $r$ is the non-zero remainder of the division of $n$ by 40; if $r=0$, then the last two digits of the number $a^{n}$ are the same as the last two digits of the number $a^{40}$. It is recommended that readers prove the last statement.