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A positive integer can be written as a sum of $18$ of its positive divisors (not necessarily distinct) as well as a sum of $19$ positive divisors. Prove that it can also be written as a sum of $20$ of its positive divisors.

We can write $n=d_1+d_2+\dots+d_{18}=e_1+e_2+\dots+e_{19}$ where the $d_i$'s and $e_i$'s all divide $n$. Maybe we can consider the parity of the divisors, but if $n$ is even then the divisors can be either even or odd. On the other hand, if $n$ is odd then all divisors are odd, so the equation above is impossible.

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As you wrote, $n$ is even.

Suppose that $e_i$ is odd for all $i$. Then, $n$ is the sum of 19 odd numbers from which we know that $n$ is odd. This is a contradiction.

So, we may assume that $e_1=2m$ where $m\in\mathbb Z$. Now, we can have $$n=m+m+e_2+e_3+\cdots+e_{19}.$$

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