Problem 20 · 1987 AJHSME
Hard
Number Theory
counterexample
"If a whole number n is not prime, then the whole number n − 2 is not prime." A value of n which shows this statement to be false is
Show answer
Answer: A — 9.
Show hints
Hint 1 of 2
To break an 'if ... then ...' rule you need ONE case where the 'if' part is true but the 'then' part fails. What must n and n − 2 each be?
Still stuck? Show hint 2 →
Hint 2 of 2
You need n itself NOT prime (so the 'if' holds) yet n − 2 prime (so the 'then' fails). Scan the choices for a composite n whose n − 2 is prime.
Show solution
Approach: make the hypothesis true but the conclusion false
- A counterexample needs the 'if' satisfied and the 'then' broken: n must be non-prime, while n − 2 must BE prime.
- n = 9 fits perfectly: 9 = 3 × 3 is not prime, yet 9 − 2 = 7 is prime. The rule predicted 7 would be non-prime, so the rule is false. Answer 9.
- Why this transfers: to disprove any 'if P then Q,' you only ever need a single example with P true and Q false — never a general argument. Choices like 13 (prime, so 'if' fails) can't be counterexamples at all.
Mark:
· log in to save