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.

Your rules are not defined  clearly enough by the  examples
provided by you:

a. "If x gives y, then 2x gives yy."  Does it mean that 3x gives yyy?.
What does xx give?

b. "213 gives 33 since 13 gives 3."   True, if a priority is given to apply the conversion rules from right to left;
if not  213=2(13)=1313=
1(313)=313=3(13)=131313=....3(1313)=131313131313=....diverges

Please  clarify the above issues.
Maybe a longer sample chain of transformations is needed.

Edited on December 2, 2014, 8:26 am

Posted by Ady TZIDON on 2014-12-02 08:06:54