TITLE: For which number does multiplying it by 99 add a 1 to each end of its decimal representation?
QUESTION [34 upvotes]: This was asked by my maths lecturer a couple of years ago and ive been wracking my brains ever since:

Find a number that, when multiplied by
  99 will give the original number but
  with a 1 at the beginning and a 1 at
  the end.
For example: 42546254 * 99 would equal
  1425462541 (it doesn't, but it
  illustrates what the answer would look
  like)

REPLY [10 votes]: It was shown that $\rm\ x = 112359550561797752809\:.$
Notice that $1/89\ =\ 0.0112359550561797752808988\ldots$
EXERCISE $\: $ Explain it (this, perhaps, is the point of the OP).
NOTE $\ $ This is closely connected with fibonacci numbers. Hint:  
$\rm\quad\quad\quad x^n\ =\ f_n\ x + f_{n-1}\ (mod\ x^2-x-1)\ $  
and note $\rm\ f_{11} = 89\ $ which is $\rm\ x^2-x-1\ $ for $\rm\ x = 10\:.$