Useless Clocks

> "Even a broken clock is right twice a day" For those unfamiliar with the above proverb, the statement refers to a stopped 12 hour clock. The time it shows happens to coincide with the true time twice every 24 hours. Chef has a useless clock which does not run properly. More specifically, it runs at $K$ times the rate of a correct clock, where $K \ne 1$. It may even be that $K$ is negative,