Here are some hints to get you started:

3: Write $\dfrac{a}{b} + \dfrac{14b}{9a}$ as a single fraction. What do you need to ask of $a$ and $b$ so that the resulting fraction is actually an integer?

9:  Start by looking at the ones digits of $N$ in base $10$, $5$, and $6$. For example, if the last digit of $N$ is base $10$ is $7$, the last digit of $N$ in base $5$ is $2$, and the last digit of $2N$ in base $10$ is $4$, thus for $S$ to have the same last digit as $2N$, the last digit of $N$ in base $6$ must be $4 - 2 = 2$.  (Note a digit in a number expressed in base $5$ is at most $4$, and in base $6$ is at most $5$, so there will not be any carryover when these are added as if they were numbers in base $10$.)

10: Find a pattern for the remainders of powers of $3$ when divided by $40$.