I have a math PhD. I got interviewed at a community college today for a teaching position. One question was to find the last digit of 3\^(large number). I immediately know it's mod 10. But my brain went completely wild. Fermat? No, 10 is not prime. Euler then. Wait, what is the formula for phi(n) again? Eh, forget it, maybe split into mod 2 and mod 5? Before I know it, time was up and I turned in blank paper. It was immediately after interview that I realized I could have easily computed 3\^4=81.
Don't be like me.