16. Are there 1989 positive integers $a_{1}, a_{2}, \ldots, a_{1989}$ such that for every $i=3,4, \ldots, 1989$ we have

$$
a_{i}+S_{i}=\left(a_{i}, S_{i}\right)+\left[a_{i}, S_{i}\right]
$$

where $S_{i}=\sum_{j=1}^{i} a_{j}$ and $(\cdot, \cdot)$ and $[\cdot, \cdot]$ are the greatest common divisor and the lowest common multiple respectively.
16. Let $x_{i}=\left(a_{i}, S_{i}\right)$ and $y_{i}=\left[a_{i}, S_{i}\right]$. Then $x_{i} y_{i}=a_{i} S_{i}(i=3,4, \ldots, 1989)$ and the given relation can be written in the form

$$
a_{i}+S_{i}=x_{i}+\frac{a_{i} S_{i}}{x_{i}}
$$

i.e. $x_{i}^{2}-\left(a_{i}+S_{i}\right) x_{i}+a_{i} S_{i}=0$. So if $a_{i} \mid S_{i}$, then $x_{i}=a_{i}$ and the equation holds true.

Thus we may take $a_{1}=1, a_{2}=2$ and $a_{i}=2^{i-3} \cdot 3, i \geq 3$, whereupon we have $S_{i}=3 \cdot 2^{i-2}, i=3,4, \ldots, 1989$.