A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, at least $2$ of any $40$ gave exactly $2$ correct answers, and at least $3$ of any $40$ gave exactly $1$ correct answer. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?

Let's think step by step and output the final answer within \boxed{}.
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