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 1 (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
Solution 2
The equation given in the problem can be written as
A counter example is and
.
Thus,
.
First, we prove the ``if part.
Suppose and
. However,
.
Thus, must be divisible by at least one factor of 210. W.L.O.G, we assume
is divisible by 2.
Modulo 2 on Equation (1), we get that .
This is a contradiction with the condition that
.
Therefore, the ``if part in Statement III is correct.
Second, we prove the ``only if part.
Suppose . Because
, there must be one factor of 14 or 15 that divides
.
W.L.O.G, we assume there is a factor
of 14 that divides
.
Because
, we have
.
Modulo
on Equation (1), we have
.
Because , we have
.
Analogously, we can prove that .
This is simply a special case of the ``only if part of Statement III. So we omit the proof.
All analysis above imply
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)