2023 AMC 10B Problems/Problem 18
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, 210) = 1.
II. If gcd(𝑐, 210) = 1, then gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both.
III. gcd(𝑐, 210) = 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