I can only try-
“Dr. Pepper – don’t be modest now, because I’m really curious – is this level of math expertise common among math majors? Among holders of a math PhD?”
I hate questions like this because I don’t want to appear arrogant. The math used to solve that riddle was basic algebra and arithmetic (the MOD(x,y) function is just a fancy way of getting the remainder of x divided by y).
Let me try to explain the thought process of a mathematician the way I heard it from the son of a math professor (it’s not supposed to be funny so don’t complain that you find it corny):
Q. What do you do if you see a piece of wood on fire and a pail of water next to it?
A. Use the pail of water to put out the fire.
Q. What do you do if you see a piece of wood on fire and there is no pail of water next to it?
A. Go get a pail of water and put out the fire.
Q. What do you do if you have a pail of water?
A. Go find a piece of wood to set on fire so you can pour the bucket of water on it.
The way a mathematician goes about solving a problem is to take it apart and change it around to fit into an equation he knows how to handle.
By the time a math major gets to upper level math courses, even calculus should come naturally. When they are trying to set up a mathematical model to solve some sort of equation on an exam they don’t have time to try to figure out how to differentiate or integrate the equation.