Problem 20 · 2015 Math Kangaroo
Medium
Number Theory
primescasework
Which value of the variable n is a counterexample to the statement “If n is a prime number, then exactly one of the two numbers n − 2 and n + 2 is a prime number.”?
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Answer: E — 37
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Hint 1 of 2
You want a prime n for which the rule fails: both n−2 and n+2 prime, or neither.
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Hint 2 of 2
Check each option: 37 is prime but 35 and 39 are both composite, so neither n−2 nor n+2 is prime.
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Approach: find the prime that breaks the claim
- The statement fails if, for a prime n, the count of primes among n−2, n+2 is 0 or 2.
- For 37 (prime): 35 = 5·7 and 39 = 3·13 are both composite → neither is prime.
- So 37 is the counterexample.
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