Here are some hints.

5.19: If a number $a$ is divisible by $k$, then you can write $a=xk$ where x is an integer. Bezout's identity is not required for this one.

5.20: To find out about 169, probably you can start with 13.

5.21 is okay, write it clearly in your homework.

5.22: There are at least two approaches that can work. First method: when powers of a number is involved in divisibility or remainder questions, Fermat's Little Theorem is often useful.  Note that $30=2\times3\times5$.  Second method: can you factor $a^5-a$, and then use the same method as in 5.21?

5.23: Consider pairing.

5.24: Do not just memorizing the rules without understanding why.  The divisibility rule for $13$ in the textbook is based on the fact that $13$ is a factor of $1001$.  Consider pairing again.