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Estonia number theory
Problem
Does there exist a positive integer such that across all representations , the digits are all present as the final digit of at least once?
Solution
Answer: No.
If is odd, both and will always be odd, but then will never end with a . If is even, then or will be even, so will be even or a perfect square. In either case, it will never end in a .
If is odd, both and will always be odd, but then will never end with a . If is even, then or will be even, so will be even or a perfect square. In either case, it will never end in a .
Final answer
No
Techniques
Modular ArithmeticIntegers