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2000 AMC 10 Problem 11

Problem 11 of 25IntermediateNumber TheoryProblem-Solving Techniques

Two different prime numbers between 44 and 1818 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

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Solution

The primes between 44 and 1818 are 5,5, 7,7, 11,11, 13,13, and 17.17. The product of two of them is odd and the sum is even, so xy−(x+y)xy - (x + y) is odd. Since xy−(x+y)xy - (x + y) =(x−1)(y−1)−1= (x - 1)(y - 1) - 1 increases as either prime increases, the result ranges from 5⋅7−12=235 \cdot 7 - 12 = 23 up to 13⋅17−30=191.13 \cdot 17 - 30 = 191. The only odd option in [23,191][23, 191] is 119=11⋅13−(11+13).119 = 11 \cdot 13 - (11 + 13). Thus, the correct answer is C.
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Tagged: prime · parity · bounding to limit cases

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