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Retrieval Quiz

Attempt closed-book, then check the answer key.

  1. Why is the canonical remainder of -23 modulo 7 equal to 5 rather than -2?
  2. Does a valid Bezout identity alone prove that the reported positive value is the gcd? What else must be checked?
  3. Why is 1 not prime, and why must a trial-division loop include the square-root boundary?
  4. Give the exact condition for an inverse modulo m>1 and explain failed cancellation modulo 6.
  5. State the invariant for repeated squaring. What does the exponent-zero/modulus-one case return?
  6. When does ax ≡ b (mod m) have solutions, and how many canonical representatives does it have when solvable?

The quiz identifies repairs; progression is decided by the assessment gates, including independent transfer.