Find the distance of each step from the ground.

$9$th step on the first ladder and $13$th step on the second ladder.

On the $k$th step of the first ladder, the distance from the ground is $\frac{20}{34}\cdot k$ feet.

and on the $m$th step of the first ladder, the distance from the ground is $\frac{20}{49}\cdot m$ feet.

The distance between them is $| \frac{20}{34}\cdot k-\frac{20}{49}\cdot m|=\left(\frac{20}{34\cdot 49}\right)|\left(49k-34m\right)|$.

This is minimized if $|\left(49k-34m\right)|=1$, which is true when $k=9$ and $m=13$.