Problem 4 · 2014 AMC 8
Easy
Number Theory
parityprimes
The sum of two prime numbers is 85. What is the product of these two prime numbers?
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Answer: E — 166.
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Hint 1 of 2
85 is odd. Two odds add to an even and two evens add to an even — so an odd total forces one even number and one odd number.
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Hint 2 of 2
Among all primes only 2 is even, so one prime is pinned down instantly. No searching required.
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Approach: an odd sum forces one even addend, and 2 is the only even prime
- 85 is odd. Odd + odd = even and even + even = even, so to get an odd sum exactly one prime must be even.
- The only even prime is 2, so that prime is 2; the other is 85 − 2 = 83 (prime — check).
- Product: 2 × 83 = 166.
- Why this transfers: "even prime" almost always means just "2". Watch the parity of a sum and you can often nail one number before doing any real work.
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