I have a number machine. I say x gives y if when x goes in, y comes out. For any numbers x and y, by "xy" I mean x followed by y. Here are my two rules.

1x gives x.
For example, 13 gives 3.
If x gives y, then 2x gives yy.
For example, 213 gives 33 since 13 gives 3.

Show that for any number a, there is a number x such that x gives ax.