Solve this alphametic, where each capital letter in bold represents a different decimal digit from 0 to 9. None of the numbers can admit any leading zero.

                                 (NOW)^THE = (OLD)^HAT

First consider any prime p.  It is either coprime to both NOW and OLD or it is a factor of both.  The equation can then be written as (p^a * q1)^THE = (p^b * q2)^HAT with q1 and q2 coprime to p.  Then a/b = HAT/THE.

This applies to every prime factor p, so then n can be defined as p1^a1 * p2^a2 * ... * pn^an.  Then the equation can be expressed as the following system:
n^a = NOW
n^b = OLD
a/b = HAT/THE

NOW and OLD are different three digit perfect powers of some number n.  n can be 2, with powers 128, 256, 512; 3 with powers 243, 729; or 5 with powers 125, 625

There are only two pairs which fit the digit pattern described by NOW and OLD: {128, 256} and {729, 243}.  Only for the second pair can the alphametic be solved for: 729 ^ 680 = 243 ^ 816

Posted by Brian Smith on 2009-02-21 18:49:40