2004 Pan African MO Problems/Problem 1
Revision as of 16:50, 24 March 2020 by Rockmanex3 (talk | contribs) (Solution to Problem 1 -- number theory casework)
Problem
Do there exist positive integers and
such that:
Solution
Note that can only be congruent to
modulo 9. So we can check the remainder when
is divided by 9. There are only nine cases to check, so we can do it manually. To make computation easier, we can factor the left-hand side as
.
![]() |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
![]() |
7 | 4 | 7 | 7 | 4 | 7 | 7 | 4 | 7 |
There are no cases of where
or
, so there are no positive integers
that satisfies the original equation.
See Also
2004 Pan African MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All Pan African MO Problems and Solutions |