Quadratic Residues and the Law of Reciprocity · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Which remainders are squares

Mathematics · Number Theory · ages 21-22
Name ______________________   Date ____________

  1. How many nonzero quadratic residues exist modulo 7?

    Answer: ______________

  2. Modulo 5, which pair lists all nonzero squares?

    • {1, 2}
    • {2, 3}
    • {1, 4}
    • {1, 3}

  3. Modulo 5, which pair lists all nonzero squares?

    • {1, 2}
    • {1, 4}
    • {2, 3}

  4. Modulo 7, the number 2 is a square because 3 squared leaves remainder 2. True or false?

    Circle one:   True   False

  5. For an odd prime p, the residue count is p minus 1 over 2. What does that give for p equal 7?

    Answer: ______________

  6. Euler criterion says an odd prime not dividing a gives remainder 1 or prime minus 1 when computing a to the power half of prime minus 1. Remainder 1 means residue, prime minus 1 means nonresidue. Is this statement correct?

    Circle one:   True   False

  7. Modulo 7, 2 raised to the power 3 leaves remainder 1. What does Euler criterion say about 2?

    • 2 is a quadratic residue
    • 2 is a quadratic nonresidue
    • Criterion gives no info
    • 2 is divisible by 7

  8. Modulo 7, 2 raised to the power 3 leaves remainder 1. What does Euler criterion say about 2?

    • 2 is a quadratic nonresidue
    • 2 is a quadratic residue
    • The criterion gives no answer here

  9. Without computing powers, why is 2 a square modulo 7?

    • 7 leaves 7 mod 8, making 2 a square
    • 7 is odd, making every number a square
    • 7 is prime, making 2 a nonsquare

  10. Consider odd primes below 15, excluding 3 itself. For how many of them is 3 a quadratic residue?

    Answer: ______________

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Answer key

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Which remainders are squares W1-mt_0KS9eTCY4p-s1

  1. 3 · Nonzero squares modulo 7 are 1, 2, and 4, so the count is 3.
  2. {1, 4} · {1, 4} is correct since 1 squared and 4 squared give 1, while 2 squared and 3 squared give 4, covering all nonzero remainders.
  3. {1, 4} · Squaring 1, 2, 3, 4 leaves 1, 4, 4, 1, so only 1 and 4 appear.
  4. True · Three squared is 9, and 9 minus 7 leaves 2.
  5. 3 · Half of 7 minus 1 is 3, giving 3 residues.
  6. True · This is exactly Euler criterion, distinguishing squares from nonsquares by the two possible remainders.
  7. 2 is a quadratic residue · 2 is a quadratic residue since Euler criterion links remainder 1 to being a square.
  8. 2 is a quadratic residue · Remainder 1 from that power signals a square.
  9. 7 leaves 7 mod 8, making 2 a square · The supplementary law puts 2 among squares exactly for primes near 1 or 7 mod 8.
  10. 2 · Eligible primes are 5, 7, 11, 13. Residues occur at 11 and 13, so the count is 2.
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