2022 Cayley Problem 25

There are T tokens arranged in a circle for some positive integer T. Moving clockwise around the circle, the tokens are labelled, in order, with the integers from 1 to T. Starting from the token labelled 1, Évariste:

(i) Removes the token at the current position.

(ii) Moves clockwise to the next remaining token.

(iii) Moves clockwise again to the next remaining token.

(iv) Repeats steps (i) to (iii) until only one token remains.

When T = 337, the number on the last remaining token is L. There are other integers T for which the number on the last remaining token is also L. What are the rightmost two digits of the smallest possible value of T?

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.