You have a simple (base-ten, whole number) calculator which can perform only two operations: visually reversing a number, and adding seven.

Prove that you can use this calcluator to convert any number to 1.

Notation: use ~ to denote reversal, as in
~53 = 35

Let the given number be n and suppose first that n is not a multiple of 7. Since the powers of 10 repeatedly cycle through 1,3,2,6,4,5,... mod 7, there are always nonnegative k and m such that n+7k=10^m which then reverses to 1.

If n is a multiple of 7, then there is a nonnegative k such that n+7k=77...7 so that n+7(k-2)=77...763 . Reversing, 3677...7 cannot be a multiple of 7, however. For 36 is congruent to 1 mod 7, while 10^m is never congruent to 0 mod 7, making it impossible for 36*10^m+77...7 to be congruent to 0 mod 7.

Posted by Richard on 2004-02-23 20:41:28