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**