2002 Pan African MO Problems/Problem 3
Problem
Prove for every integer , there exists an integer such that can be written in decimal notation using only digits 1 and 2.
Solution
We can use induction to solve the problem. For the base case, note that is divisible by , is divisible by , is divisible by , and is divisible by .
Now assume that . That means , where is an integer. Multiply both sides by to get , so . That means or .
Additionally, from above, note that and . Thus, if , then , and if , then . Therefore, it is possible to write a number with only digits 1 and 2 that is divisible by , so there exists a positive integer where can be written in decimal notation using only digits 1 and 2.
See Also
2002 Pan African MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All Pan African MO Problems and Solutions |