Problem 18 · 2015 Math Kangaroo
Medium
Algebra & Patterns
primesdigit-sum
It is known that the solutions of the quadratic equation \(x^2 - 85x + c = 0\) are prime numbers. What is the digit sum of c?
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Answer: B — 13
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Hint 1 of 2
The two roots add to 85 (an odd number), so one root must be the only even prime.
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Hint 2 of 2
If one root is 2, the other is 83; then c is their product — take its digit sum.
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Approach: use sum of roots = 85, both prime
- By Vieta, the roots sum to 85 (odd), so one is even → it must be 2.
- The other root is 85 − 2 = 83, which is prime.
- c = product = 2 × 83 = 166; digit sum = 1 + 6 + 6 = 13.
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