Constructing DFAs $(30$ points; each part $10$ points$)$

For each of the following languages over the alphabet $=\{0,1\},$ give a DFA that recognizes the language, and explain why it's correct. Try to use as few states as possible.

a$.$ has an odd number of $1s.$ For instance, $11111$ has five $1s$ and is in the language, and $001000$ has one $1$ and is in the language, while $11001010$ has four $1s$ and is not in the language.

b$.|)$:$\}$ starts and ends with the same character