Numbers have different representations depending on the
bases on which they are expressed.
For example, in base 3, number 12 is written as 110 but
in base 8 it is written as 14.
For this problem your program will presented with a sequence
of pairs of integers. Let’s call the members of a pair X and Y.
What your program is to do is determine the smallest
base for X and the smallest base for Y (likely different from that for
X) so that X and Y represent the same value.
Consider, for example, the integers 12 and 5. Certainly
these are not equal if base 10 is used for each. But suppose 12 was a base
3 number and 5 was a base 6 number?
12 base 3 = 1 × 3 + 2 × 1, or 5 base 10,
and certainly 5 in any base is equal to 5 base 10. So 12 and 5 can be equal,
if you select the right bases for each of them!
Input:
On each line of the input data there will be a pair of
integers, X and Y, separated by one or more blanks; leading and trailing
blanks may also appear on each line, are are to be ignored.
The bases associated with X and Y will be between 1 and
36 (inclusive), and as noted above, need not be the same for X and Y.
In representing these numbers the digits 0 through 9
have their usual decimal interpretations.
The uppercase alphabetic characters A through Z represent
digits with values 10 through 35, respectively.
The last line of the input will contain nothing but zero
or more blanks (and an end of line, of course), and represents the end
of the data.
There will be no incorrectly formatted data in the input.
Output:
For each pair of integers in the input display a message
similar to those shown in the examples shown below.
Of course if the two integers cannot be equal regardless
of the assumed base for each, then print an appropriate message; a suitable
illustration is given in the examples.
Sample Input:
12 5
10 A
12 34
123 456
1 2
10 2
Sample Output:
12 (base 3) = 5 (base 6)
10 (base 10) = A (base 11)
12 (base 17) = 34 (base 5)
123 is not equal to 456 in any base 2..36
1 is not equal to 2 in any base 2..36
10 (base 2) = 2 (base 3)