If you moved the 3 at the end of a certain large number to the beginning, it would be the same as multiplying it by 3. What is the number?

There's a standard way of solving this kind of "move-to-the-beginning" problems. Assume the solution is a number abc...3 and let X be the continued fraction 0.abc...3abc...3abc...3...

Then, (X+3)/10= 0.3abc...3abc...3abc..., which must be equal to 3X.

Equating, we get X=3/29, which works out to be 0.1034482758620689655172413793..., so our number is 1034482758620689655172413793.
Edited on March 1, 2004, 12:01 pm

Posted by Federico Kereki on 2004-03-01 12:00:48