Primitive Roots and the Units Modulo n · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Finding a generator for the units

Mathematics · Number Theory · ages 20-21
Name ______________________   Date ____________

  1. What is the order of 1 modulo 17?

    Answer: ______________

  2. What is the order of 2 modulo 5? That is, the smallest positive k with 2 to the k equal to 1 mod 5.

    Answer: ______________

  3. What is the order of 2 modulo 7?

    • 6
    • 3
    • 2

  4. Leo says 4 can be a primitive root modulo 10 even though 4 and 10 share a factor. Is this true?

    Circle one:   True   False

  5. The primitive-root classification says that a primitive root exists modulo 18. Is this true?

    Circle one:   True   False

  6. Mia says every modulus has a primitive root.

    Circle one:   True   False

  7. What is the order of 4 modulo 9?

    Answer: ______________

  8. A unit has order 4 modulo 15. Which conclusion follows?

    • phi(15) = 4; the unit is primitive
    • phi(15) = 8; the unit is not primitive
    • phi(15) = 14; the unit is primitive
    • phi(15) = 15; the unit is not primitive

  9. Modulo 13, 2^6 gives 12 and 2^4 gives 3. Since phi(13) = 12, what does the prime-divisor test show?

    • 2 has order 6, so is not primitive
    • 2 has order 4, so is not primitive
    • 2 has order 12, so is primitive

  10. 2 is primitive modulo 11, and 2^4 is congruent to 5. What is the order of 5 modulo 11?

    • 5
    • 4
    • 10
    • 2
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Answer key

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

Finding a generator for the units W1-mt_KwTDckkMze-s1

  1. 1 · The first positive power, 1^1, already gives remainder 1. Its order is 1, not 0.
  2. 4 · The powers go 2, 4, 3, 1, so 1 first returns at the 4th.
  3. 3 · The powers go 2, 4, 1, so 1 first returns at exponent 3.
  4. False · A primitive root must be a unit. Since gcd(4,10) = 2, 4 is not a unit modulo 10.
  5. True · 18 = 2 times 3^2, so it has the allowed form 2p^k with p an odd prime and k positive.
  6. False · Mod 8 has none: every odd square is 1 there while phi of 8 is 4.
  7. 3 · The successive remainders are 4, 7, 1. The first return to 1 is at exponent 3.
  8. phi(15) = 8; the unit is not primitive · There are eight units modulo 15. A primitive root would need order 8, so order 4 is insufficient.
  9. 2 has order 12, so is primitive · The prime divisors of 12 are 2 and 3. Neither required test power gives 1, so 2 has order 12 and is primitive.
  10. 5 · The order is 10/gcd(10,4) = 10/2 = 5.
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