Problem 19 · 2018 Math Kangaroo
Hard
Number Theory
caseworkprimes
Three of the cards shown will be dealt to Nadia, the rest to Riny. Nadia multiplies the three values of her cards and Riny multiplies the two values of his cards. It turns out that the sum of those two products is a prime number. Determine the sum of the values of Nadia's cards.
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Answer: B — 13
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Hint 1 of 2
For the sum of the two products to be an odd prime, one product must be even and the other odd.
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Hint 2 of 2
Riny's two-card product is odd only if both his cards are odd — try the odd pairs from 3,5,7.
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Approach: use parity, then test the few odd pairs for Riny
- The card values are 3,4,5,6,7. A prime above 2 is odd, so one product is even, one odd.
- Riny's two-card product is odd only when both his cards are odd; the odd values are 3,5,7.
- Trying Riny = {5,7}: product 35, Nadia = {3,4,6}, product 72, and 72+35 = 107, which is prime.
- Nadia's cards are 3, 4, 6, summing to 13.
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