For how many integers $a$ satisfying $1 \le a \le 23$ is it true that $a^{-1} \equiv a \pmod{24}$?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
For how many integers $a$ satisfying $1 \le a \le 23$ is it true that $a^{-1} \equiv a \pmod{24}$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
For how many integers $a$ satisfying $1 \le a \le 23$ is it true that $a^{-1} \equiv a \pmod{24}$?
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