Proof by Contradiction and Counterexample · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One counterexample, one impossible ending

Mathematics · Mathematical Thinking · ages 17-18
Name ______________________   Date ____________

  1. What is the smallest prime number?

    Answer: ______________

  2. Which number disproves the claim that every prime number is odd?

    • 9
    • 2
    • 1
    • 15

  3. Which number disproves the claim that every prime number is odd?

    • 2
    • 9
    • 15

  4. A proof by contradiction opens by assuming the claim is false.

    Circle one:   True   False

  5. The rule f(n) = n squared + n + 41 gives primes for n = 0 to 39. What is f(40)?

    Answer: ______________

  6. The number 1681 equals which square?

    • 40 squared
    • 41 squared
    • 41 times 43

  7. The classic proof that root 2 is irrational starts by assuming what?

    • root 2 = a/b in lowest terms
    • root 2 has a repeating decimal
    • root 2 lies between 1 and 2
    • root 2 is negative

  8. Assuming root 2 = a/b in lowest terms forces both a and b even, collapsing the assumption. True or false?

    Circle one:   True   False

  9. Suppose root 2 is rational leads to 1 = 0 style nonsense. What do you conclude?

    • the algebra had an error
    • fractions cannot be trusted
    • root 2 is irrational

  10. Suppose root 2 is rational leads to 1 = 0 style nonsense. What do you conclude?

    • maths is broken
    • fractions cannot be trusted
    • the algebra had an error
    • root 2 is irrational
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Answer key

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

One counterexample, one impossible ending W1-mt_pbFTv4v6k8-s1

  1. 2 · 2. It has exactly two divisors and every smaller candidate fails.
  2. 2 · 2. It is prime and even, a single counterexample that sinks the for all claim.
  3. 2 · 2 is prime and even, a single counterexample that sinks the for all claim.
  4. True · The method assumes the negation and derives nonsense from it.
  5. 1681 · 1681. Note 1600 + 40 + 41 = 1681, which later factors and breaks the pattern.
  6. 41 squared · 41 squared is 1681, so f(40) is composite and breaks the always prime pattern.
  7. root 2 = a/b in lowest terms · root 2 = a/b in lowest terms. Squaring forces both a and b even, contradicting lowest terms.
  8. True · True. Even a gives a squared divisible by 4, forcing b even and contradicting lowest terms.
  9. root 2 is irrational · The only disposable assumption was rationality itself.
  10. root 2 is irrational · root 2 is irrational. The only disposable assumption was rationality itself.
Worksheet · LightMySky