Euler’s Theorem states that for a positive integer and an integer relatively prime with
where is the Euler’s function.
Problem (Baltic Way, 1997)
If we add to , we get , thus making three carries in total. Does there exist a positive integer , such that adding to no carry arises during the whole calculation?
Solution
Answer: yes, there exists such a number .
Let us prove that there exists a number that consists entirely of digits and is multiple of . This will automatically imply that if is a sum of some two numbers and , then there were no carries during the calculation.
Since and are relatively prime, then by Euler’s Theorem
Therefore
Now we can put . Note that consists entirely of digits and also
Now for the value of we can take the number .