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Is it true, that odd numbers have only odd divisors? If yes, what would a formal proof look like?

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Yes. If $a=2k\vert b$ then there is $d$ such that $b=2kd$, which means $b$ is even.

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A number $n$ is odd if $2$ does not divide $n$.

Let $d$ be a divisor of $n$. If $d$ is not odd, then $2\mid d$. But then $2\mid n$ too, so that $n$ cannot be odd as well.

Then odd numbers cannot have divisors that are not odd.

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You can use the contrapositive:

If a number has an even divisor then it is even.

This is easily proved because "to divide" is transitive: $$ 2 \mid d \mid n \implies 2 \mid n $$

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