6. Viktor thought of a number. He added the thought number to 456. He continued to add the thought number to each obtained sum until he got the number 555. He repeated the process of adding the thought number more than 10 times. What number did Viktor think of? Determine all possibilities and write the explanation.
First solution.

$555-456=99$

The number 99 needs to be written as the sum of equal addends. 1 POINT

If the number 1 was guessed, then that number was added $99: 1=99$ times. 1 POINT

If the number 2 was guessed, then that number would be added $99: 2$ times, which is impossible. 1 POINT

Similarly, he could not have guessed any other even number because the number 99 is odd. 1 POINT

If the number 3 was guessed, then that number was added $99: 3=33$ times. 1 POINT

If the number 5 was guessed, then that number would be added $99: 5$ times, which is impossible. 1 POINT

If the number 7 was guessed, then that number would be added $99: 7$ times, which is impossible. 1 POINT

If the number 9 was guessed, then that number was added $99: 9=11$ times. 1 POINT

If the number 11 was guessed, then that number was added $99: 11=9$ times. 1 POINT

If we continue to increase the number that Viktor guessed, then the number of additions will continue to decrease. This means that the guessed number will be added fewer than 10 times, which is not possible due to the conditions of the problem.

1 POINT

TOTAL 10 POINTS

Note: To eliminate the possibility that the guessed number is even, an explanation must be written. Otherwise, the student must eliminate all remaining possibilities (the guessed number cannot be 4, 6, 8, and 10) for 1 POINT.