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| − | == Problem ==
| + | #redirect[[2023 AMC 12B Problems/Problem 15]] |
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| − | Suppose 𝑎, 𝑏, and 𝑐 are positive integers such that
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| − | <math>\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}</math>.
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| − | Which of the following statements are necessarily true?
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| − | I. If gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both, then gcd(𝑐, 21) = 1.
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| − | II. If gcd(𝑐, 21) = 1, then gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both.
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| − | III. gcd(𝑐, 21) = 1 if and only if gcd(𝑎, 14) = gcd(𝑏, 15) = 1.
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| − | == Solution (Guess and check + Contrapositive)==
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| − | <math>I.</math> Try <math>a=3,b=5 => c = 17\cdot15</math> which makes <math>\textbf{I}</math> false.
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| − | At this point, we can rule out answer A,B,C.
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| − | <math>II.</math> A => B or C. equiv. ~B AND ~C => ~A.
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| − | Let a = 14, b=15 (statisfying ~B and ~C). => C = 2*210. which is ~A.
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| − | <math>II</math> is true.
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| − | So the answer is E.
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| − | <math>\boxed{\textbf{(E) } II \text{ and } III \text{only}.}</math>
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| − | ~Technodoggo
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