2004 Pan African MO Problems/Problem 1
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 |