Difference between revisions of "2023 AMC 10B Problems/Problem 18"
Technodoggo (talk | contribs) (→Solution) |
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| + | == Problem == | ||
| + | |||
| + | Suppose 𝑎, 𝑏, and 𝑐 are positive integers such that | ||
| + | <math>\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}</math>. | ||
| + | |||
| + | Which of the following statements are necessarily true? | ||
| + | I. If gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both, then gcd(𝑐, 21) = 1. | ||
| + | II. If gcd(𝑐, 21) = 1, then gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both. | ||
| + | III. gcd(𝑐, 21) = 1 if and only if gcd(𝑎, 14) = gcd(𝑏, 15) = 1. | ||
| + | |||
== Solution (Guess and check + Contrapositive)== | == Solution (Guess and check + Contrapositive)== | ||
<math>I.</math> Try <math>a=3,b=5 => c = 17\cdot15</math> which makes <math>\textbf{I}</math> false. | <math>I.</math> Try <math>a=3,b=5 => c = 17\cdot15</math> which makes <math>\textbf{I}</math> false. | ||
Revision as of 16:43, 15 November 2023
Problem
Suppose 𝑎, 𝑏, and 𝑐 are positive integers such that
.
Which of the following statements are necessarily true? I. If gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both, then gcd(𝑐, 21) = 1. II. If gcd(𝑐, 21) = 1, then gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both. III. gcd(𝑐, 21) = 1 if and only if gcd(𝑎, 14) = gcd(𝑏, 15) = 1.
Solution (Guess and check + Contrapositive)
Try 
which makes 
false.
At this point, we can rule out answer A,B,C.
A => B or C. equiv. ~B AND ~C => ~A.
Let a = 14, b=15 (statisfying ~B and ~C). => C = 2*210. which is ~A.
is true.
So the answer is E.
~Technodoggo
