Problem

Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p6

Tags: Mexico



Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations: $1.$ Erase the written number $m$ and write it´s cube $m^3$. $2.$ Erase the written number $m$ and write the product $2023m$. Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.