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?

(In reply to Standard solution by Federico Kereki)

I am fascinated by this non-practical problem and the method you give.  Can you give a reference for the method?  Replacing 3 in the problem with 2, I get the 18-digit number 105263157894736842. Replacing 3 with n, there will be an answer (perhaps itself containing an internal repeating answer) with 10n-2 digits if 10n-1 is prime, i.e. when n=2,3,6,8, and 9.  When 10n-1 is prime, the (10n-2)-digit answer will be the smallest possible whenever the index of 10 mod 10n-1 is coprime to 10n-2 as it is for n=2,3, and 6. Thus the minimal answer for n=6 has 58 digits.

Posted by Richard on 2004-03-02 01:10:08