Problem 16 · 1989 AJHSME
Hard
Number Theory
parityprimes
In how many ways can 47 be written as the sum of two primes?
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Answer: A — 0.
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Hint 1 of 3
47 is odd. Two numbers add to an odd total only when one is even and one is odd — so think about which of the two primes is the even one.
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Hint 2 of 3
Use parity to shrink the search: an odd sum forces one even prime, and there's only a single even prime in existence.
Still stuck? Show hint 3 →
Hint 3 of 3
The only even prime is 2, so the partner is forced to be 47 − 2 = 45. Now just check: is 45 prime?
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Approach: parity forces one prime to be 2
- Two numbers add to an odd total (47) only if one is even and the other odd. Among primes, the lone even one is 2 — every other prime is odd. So if 47 splits into two primes, one of them MUST be 2.
- That forces the partner to be 47 − 2 = 45. But 45 = 5 × 9 is not prime, so no valid split exists: 0 ways.
- Why this transfers: a parity check often collapses a search instantly. 'Odd = even + odd' plus 'the only even prime is 2' means an odd number is a sum of two primes only if (number − 2) is itself prime — one test, done. Try it on 13: 13 − 2 = 11 is prime, so 13 = 2 + 11 works.
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