Let $m, n$ be such divisors of $k$ and $k+1$ respectively. If $m \neq k$ and $n \neq k-1$, then $k + 1 < a = m + n \leq \frac{k}{2} + \frac{k-1}{2} < k$, contradiction, so either $m = k$ or $n = k - 1$. Now if $m = k$ then $n > 1$ and $n \mid (k - 1) + n = a - 1$. If $n = k - 1$ then $m > 1$ and $m \mid k + m = a + 1$. Done.