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Modular arithmetic practice

Modular arithmetic is arithmetic with remainders. It turns giant calculations into small ones.

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How it works

We write a ≡ b (mod m) when a and b leave the same remainder when divided by m. Clock time is mod 12: 3 hours after 11 o'clock is 2 o'clock.

The big idea: you may replace any number by its remainder before adding, subtracting or multiplying. To find 123 × 456 mod 7, reduce first (123 ≡ 4 and 456 ≡ 1), so the answer is 4 × 1 = 4. No big multiplication needed.

For powers, find a small power that is ≡ 1, then use it to cut down the exponent. Since 2³ = 8 ≡ 1 (mod 7), powers of 2 mod 7 repeat every 3. Negative remainders help too: 7 ≡ −1 (mod 8), so 7even ≡ 1 and 7odd ≡ 7.

Modular Arithmetic practice problems with solutions

These problems come from Mathyessy's question bank. Try each one before opening the solution.

  1. What is 23 times 17 mod 11?

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    Answer: 6

    Reduce first: 23 ≡ 1 and 17 ≡ 6 (mod 11). So the product is ≡ 1 × 6 = 6. (Check: 391 = 35 × 11 + 6.)

  2. What is 2 to the power 100 mod 7?

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    Answer: 2

    2³ = 8 ≡ 1 (mod 7), so the powers repeat every 3. 100 = 3 × 33 + 1, so 2¹⁰⁰ = (2³)³³ × 2 ≡ 1 × 2 = 2.

  3. What is the smallest positive x so that 7 times x is 1 mod 11?

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    Answer: 8

    Try multiples of 7 until one is 1 more than a multiple of 11: 7, 14, 21, 28, 35, 42, 49, 56. 56 = 55 + 1, so x = 8.

  4. What is 123 times 456 mod 7?

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    Answer: 4

    123 = 119 + 4, so 123 ≡ 4. 456 = 455 + 1, so 456 ≡ 1. The product is ≡ 4 × 1 = 4.

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