What quantity did you change?
Labs / Mathematics
Modular Arithmetic
The numbers 0 to m − 1 sit on a circular clock. Press play and the hand hops around by k each tick — adding or multiplying — drawing chords that trace the pattern of the operation. Tap any number to start the walk somewhere new, and drag the sliders to change the modulus and the step.
Labs / Mathematics / Modular Arithmetic
Modular Arithmetic
The numbers 0 to m − 1 sit on a circular clock. Press play and the hand hops around by k each tick — adding or multiplying — drawing chords that trace the pattern of the operation. Tap any number to start the walk somewhere new, and drag the sliders to change the modulus and the step.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
step 0residue 0cycle 12/12gcd 1 ✓ all
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- With m = 12, which values of k eventually visit every number on the clock, and which get stuck in a short loop?
- The readout shows gcd(k, m). What do all the k that reach every number have in common with m?
- Switch to “Multiply k”. Why does the pattern suddenly look like a flower or a cardioid instead of an even star?
- Set m to a prime like 7 or 13. Does that make it easier or harder to find a k that hits everything?
- Multiply by k when k shares a factor with m — where does the pointer get trapped, and can it ever return to where it started?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.