Modular Arithmetic and Congruence Classes · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Remainders you can add and multiply

Mathematics · Number Theory · ages 19-20
Name ______________________   Date ____________

  1. Work out (7 plus 9) mod 5. Give a number.

    Answer: ______________

  2. Work out (7 plus 9) mod 5. Give a number.

    Answer: ______________

  3. Kim says arithmetic mod 6 behaves exactly like ordinary integer arithmetic. She is shown that 2 times 3 is 0 mod 6. Is Kim right?

    Circle one:   True   False

  4. Work out 4 times 6 mod 7.

    • 24
    • 3
    • 4

  5. Work out 3 to the power 6 mod 7. Give a number.

    Answer: ______________

  6. What is clock arithmetic a picture of?

    • Like a 12 hour clock: 12 plus 1 wraps back to 1
    • Like a bank balance that never wraps around
    • Like a number line that runs forever

  7. Work out 2 to the power 10 mod 7 by repeated squaring. Give a number.

    Answer: ______________

  8. Work out 2 to the power 10 mod 7 by repeated squaring. Give a number.

    Answer: ______________

  9. Which of these congruences has NO integer solution?

    • x squared is 1 mod 4
    • x squared is 0 mod 4
    • x squared is 2 mod 4

  10. Which of these congruences has NO integer solution?

    • x squared is 2 mod 4
    • x squared is 1 mod 4
    • x squared is 0 mod 4
    • x squared is 1 mod 3
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Remainders you can add and multiply W1-mt_Lt4_Aazx2j-s1

  1. 1 · Adding first is legal: 7 plus 9 = 16, and 16 leaves remainder 1 on division by 5.
  2. 1 · Adding first is legal: 7 plus 9 is 16, and 16 leaves remainder 1 on division by 5.
  3. False · Kim is wrong. 2 times 3 is 6, which is 0 mod 6, even though neither factor is 0 mod 6.
  4. 3 · Multiply first: 4 times 6 is 24, and 24 is 3 times 7 plus 3.
  5. 1 · 3 cubed is 27, which is 6 mod 7 (that is, minus 1), and squaring gives 36, which is 1 mod 7.
  6. Like a 12 hour clock: 12 plus 1 wraps back to 1 · Clock arithmetic wraps after 12, which is remainders in disguise.
  7. 2 · Square twice: 2 to the 5th is 32, which is 4 mod 7, and 4 squared is 16, which is 2 mod 7.
  8. 2 · Square twice: 2 to the 5th is 32, which is 4 mod 7, and 4 squared is 16, which is 2 mod 7.
  9. x squared is 2 mod 4 · Squares mod 4 are only 0 and 1, so remainder 2 never occurs.
  10. x squared is 2 mod 4 · Squares mod 4 are only 0 and 1 (0, 1, 0, 1 for x = 0, 1, 2, 3), so remainder 2 never occurs.
Worksheet · LightMySky