Suppose there are two types of plumbers: competent and incompetent. Both types of plumbers can get certified, but the certification takes more time and effort for the incompetent types. Competent ones have to spend C months preparing for the certification exam; incompetent ones take twice as long. Certified plumbers can earn $100($thousand dollars$)$ each year working on building sites for licensed contractors. Uncertified plumbers can only earn $25($thousand dollars$)$ each year in freelance work; licensed contractors won't hire them. Each type of plumber gets a payoff equal to

$,$ where S is the salary measured in thousands of dollars and M is the number of months spent getting certified.

What is the lowest value of C for which a competent plumber will choose to signal with this certification but an incompetent one will not? $($hint: find the entire range first$).$