For 9.22 and 9.25 try to find a pattern by looking at smaller powers of the numbers. For example, The first few powers of $2$ are $2, 4, 8, 16, 32, 64, 132, \dots$ and their remainders when dividing by $15$ are $2, 4, 8, 1, 2, 4, 8,\dots$, so we can see they repeat every $4$ powers; since $102$ has reminder $2$ when dividing by $4$, the remainder of $2^{102}$ when dividing by $15$ will be the $2^\text{nd}$ remainder on the list, so $4$.

For 9.29: Recall $a \equiv b \pmod{m}$ if $m$ divides $b - a$. So, the possible values of $m$ are numbers that divide the difference $1023 - 561$.