Modular Arithmetic Tables
Modular arithmetic is the arithmetic of remainders. Fix a number $q$ and, after every calculation, keep only what is left over after dividing by $q$—so the only values that ever appear are $0, 1, \dots, q-1$. Count upward and you climb to $q-1$, then wrap straight back to $0$. Mathematicians call this working “modulo $q$”:
\[a + b \pmod{q} \qquad\text{and}\qquad a \cdot b \pmod{q}\]This applet shows the complete times table (or addition table) for a chosen $q$, with every cell shaded by its value—larger results are darker—so the arithmetic turns into a visual pattern. Click the symbol in the top-left corner to switch between the addition and multiplication table.
The two look very different. Addition simply shifts along by one at each step, giving tidy diagonal stripes. Multiplication is richer and reveals real number theory. When $q$ is a prime number, every row except the first contains each value exactly once. When $q$ is not prime, gaps and repeats appear—for instance two nonzero numbers can multiply together to give $0$—and the pattern of these surprises is tied to the divisors of $q$.
Use the slider to change $q$ from 1 to 99; the table instantly resizes and recolors, so you can flip between prime and non-prime values and watch the patterns shift.