[Math] How to explain to students that $0 \bmod n$ equals $0$

educationintuitionmodular arithmetic

I am teaching a beginning programming class in Visual Basic (for non-CS majors). I told my students that the mod operator basically gives the remainder of the division. So, when seeing $0 \bmod 10$, some students (apparently) reasoned that, "$10$ goes into $0$ zero times and there are $10$ leftover."

What would be the best way to explain this to (non-math and non-CS major) students?

I would rather counter the "$10$ goes into $0$ zero times with $10$ leftover" reasoning than change my definition, if possible. I checked the documentation for the language, and it gave the same "remainder" definition that I gave.

Best Answer

$10$ goes into $0$ zero times and there are $0$ left over.

Go back to long division the way you learned it in third or fourth grade: $$ \begin{array}{ccccccc} & & 0 \\ \\ 10 & ) & 0 \\ & & 0 \\ \hline & & 0 \end{array} $$ The remainder is $0$.

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