Question 123
A strange calculator has only two buttons with positive integers, each consisting of two digits. It displays the number $1$ at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X \cdot N$ or $X + N$. Multiplication and addition alternate, multiplication is the first. (For example, if the number $10$ is on the 1st button, the number $20$ is on the 2nd button, and we consecutively press the 1st, 2nd, 1st, and 1st button, we get the results $1 \cdot 10 = 10$, $10 + 20 = 30$, $30 \cdot 10 = 300$, and $300 + 10 = 310$.) Decide whether there exist particular values of the two-digit numbers on the buttons such that one can display infinitely many numbers ending with
(a) $2015$,
(b) $5813$.
(a) $2015$,
(b) $5813$.