Modular Arithmetic · Explore together

Find the repeating piece

Build a remainder history one power at a time, then predict a huge exponent.

One person controls the experiment. The other predicts and questions. Swap roles after a trial.

rn+1≡arn(modm)r_{n+1}\equiv a r_n\pmod m

A huge power, a small amount of information. Step through the remainders. Find where repetition begins before predicting a distant power.

70 ≡ 1 (mod 10)

Start with power 0. Multiply once to find the next remainder.

Observed remainders

    Test your cycle claim

    The loop might begin after an initial stretch. A repeated remainder gets the same next remainder because the multiplier stays fixed.

    Predict a distant power

    Explain the exponent's position in the loop to your partner. The answer stays hidden until you make a prediction.

    Compare two systems. What changes when the base shares a factor with the modulus? Try 2 mod 8, then defend why the eventual pattern continues.

    Compare your evidence

    What can you explain now?

    Ask your partner to restate your reasoning. An experiment suggests a pattern; your explanation should say why it holds and when it applies.

    1. Why does the same remainder force the following remainders to repeat?
    2. Does the cycle always start at the first power?
    3. What changes when the base and modulus share a factor?