Prove that there does not exist a natural number, which upon transfer of its initial digit to the end, is increased five, six or eight times.

The smallest digit one would be able to transfer is 1.
The largest number able to be formed would begin with 9.

Let the smallest number be:
 199......99
On transfer the number becomes:
 999......91
The smaller number will approach 1/5 of the larger but never equal it.  Since this is the limit any number greater than 5 can be disregarded.

Posted by brianjn on 2012-12-20 18:04:58