Problem 1—Double Trouble

The secret service of Hudonia has always had a strong appetite for strange
numbers. This appetite seemed strongest when Hudonia was in trouble. The
numbers they were looking for had the following property. Whenever you would
right-rotate the number (that is, take away the last digit and put it in front
of the number), you would end up with double the original number. Numbers
possessing this property were called double-trouble numbers. For example, *X*
= 421052631578947368 is a double-trouble number, since 2*X* =
842105263157894736 which is a right rotation of *X*.

The number *X* is a double-trouble number in the number system with
base 10. Any number system with base *p* ≥ 2, however, has many such
double-trouble numbers. In the binary number system (base *p* = 2), for
example, we have the double-trouble numbers 01 and 0101. Notice that the
leading zeros are necessary here in order to obtain the proper number after
right rotation.

The secret service seemed to like the smallest double-trouble numbers in a
number system most. For example, in the binary number system the smallest
double-trouble number is 01. In the decimal (*p* = 10) number system, the
smallest double-trouble number is 052631578947368421. Being a patriotic
Hudonian, you are asked to write a program that computes for a given base *p*
of a number system the smallest double-trouble number in that system.

__ __

** INPUT
SPECIFICATION.**
The input file will contain a series of integers inclusively between 2
and 200 corresponding to the base for which you are to compute the smallest
double-trouble number. The last number
in the file will be 0, signifying the end of input. Each number will be followed by

** OUTPUT SPECIFICATION.** The output cases should be processed in the
order in which they appear in the input file.
For each input case, print out the smallest double-trouble number for
that base using the exact format demonstrated below. Notice that since it is difficult to express digits for bases
greater than 10, we instead use for each digit the decimal number corresponding
to that digit followed by a space. Make
sure that the introductory line to each answer is exactly as demonstrated. Each output case should be followed by an
extra

__SAMPLE INPUT.__

2**<EOLN>**

10**<EOLN>**

35**<EOLN>**

0**<EOLN>**

**<EOF>**

__SAMPLE
OUTPUT.__

For·base·2·the·double-trouble·number·is**<EOLN>**

0·1·**<EOLN>**

**<EOLN>**

For·base·10·the·double-trouble·number·is**<EOLN>**

0·5·2·6·3·1·5·7·8·9·4·7·3·6·8·4·2·1·**<EOLN>**

**<EOLN>**

For·base·35·the·double-trouble·number·is**<EOLN>**

11·23·**<EOLN>**

**<EOLN>**

**<EOF>**