Problem 24 · 2011 AMC 8
Medium
Number Theory
parityprimes
In how many ways can 10001 be written as the sum of two primes?
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Answer: A — 0 ways.
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Hint 1 of 2
The target 10001 is odd, and odd = even + odd. Two odd primes always sum to an even number, so to hit an odd total one prime must be even — and the only even prime is 2. That leaves just one candidate pair to check.
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Hint 2 of 2
So everything hinges on a single question: is 10001 − 2 = 9999 prime?
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Approach: parity forces one prime to be 2, leaving one pair to test
- 10001 is odd. A sum is odd only when one part is even and one is odd, so one of the two primes must be even — the only even prime is 2.
- That fixes the other prime as 10001 − 2 = 9999. Check it: its digits 9+9+9+9 = 36 is divisible by 3, so 9999 = 3·3333 is not prime.
- The one possible pair fails, so there are 0 ways.
- Worth keeping: writing an odd number as prime + prime is only possible if 2 is one of them — parity collapses the whole search to a single primality check.
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