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If $x$ and $y$ are positive integers, and 1 is the greatest common div
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10 Feb 2026, 03:18
10 Feb 2026, 03:18
Expert Reply
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Question Stats:
66% (01:06) correct
33% (01:12) wrong based on 9 sessions
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If $x$ and $y$ are positive integers, and 1 is the greatest common divisor of $x$ and $y$, what is the greatest common divisor of $\(2 x\)$ and $\(3 y\)$ ?
A. Cannot be determined
B. 1
C. 2
D. 5
E. 6
A. Cannot be determined
B. 1
C. 2
D. 5
E. 6
- Part of the project: GRE Daily Quant Challenge - Topic-Wise 2025!!!
- Also replying to the unanswered questions
Re: If $x$ and $y$ are positive integers, and 1 is the greatest common div
[#permalink]
20 Feb 2026, 14:00
20 Feb 2026, 14:00
Expert Reply
The best way to see if a GCD is fixed or variable is to plug in different pairs of $x$ and $y$ that satisfy $\(\operatorname{gcd}(x, y)=1\)$.
- Case 1: Let $\(x=1\)$ and $\(y=1\)$.
- $\(\operatorname{gcd}(1,1)=1\)$ (Condition met).
- $\(2 x=2(1)=2\)$ and $\(3 y=3(1)=3\)$.
- $\(\operatorname{gcd}(2,3)=\mathbf{1}\)$.
- Case 2: Let $\(x=3\)$ and $\(y=2\)$.
- $\(\operatorname{gcd}(3,2)=1\)$ (Condition met).
- $\(2 x=2(3)=6\)$ and $\(3 y=3(2)=6\)$.
- $\(\operatorname{gcd}(6,6)=\mathbf{6}\)$.
- Case 3: Let $\(x=5\)$ and $\(y=2\)$.
- $\\(operatorname{gcd}(5,2)=1\)$ (Condition met).
- $\(2 x=2(5)=10\)$ and $\(3 y=3(2)=6\)$.
- $\(\operatorname{gcd}(10,6)=\mathbf{2}\)$.
2. Analyze the Results
Because we found three different possible values for the $\(\operatorname{GCD}(1,6\)$, and \(2\))$ while still following the rule that $\(\operatorname{gcd}(x, y)=1\)$, the value of $\(\operatorname{gcd}(2 x, 3 y)\)$ is not a single fixed number. It depends entirely on whether $x$ is a multiple of 3 and whether $y$ is a multiple of 2 .
3. Conclusion
Since the result changes depending on the specific values of $x$ and $y$ chosen, the correct answer is A. Cannot be determined.
- Case 1: Let $\(x=1\)$ and $\(y=1\)$.
- $\(\operatorname{gcd}(1,1)=1\)$ (Condition met).
- $\(2 x=2(1)=2\)$ and $\(3 y=3(1)=3\)$.
- $\(\operatorname{gcd}(2,3)=\mathbf{1}\)$.
- Case 2: Let $\(x=3\)$ and $\(y=2\)$.
- $\(\operatorname{gcd}(3,2)=1\)$ (Condition met).
- $\(2 x=2(3)=6\)$ and $\(3 y=3(2)=6\)$.
- $\(\operatorname{gcd}(6,6)=\mathbf{6}\)$.
- Case 3: Let $\(x=5\)$ and $\(y=2\)$.
- $\\(operatorname{gcd}(5,2)=1\)$ (Condition met).
- $\(2 x=2(5)=10\)$ and $\(3 y=3(2)=6\)$.
- $\(\operatorname{gcd}(10,6)=\mathbf{2}\)$.
2. Analyze the Results
Because we found three different possible values for the $\(\operatorname{GCD}(1,6\)$, and \(2\))$ while still following the rule that $\(\operatorname{gcd}(x, y)=1\)$, the value of $\(\operatorname{gcd}(2 x, 3 y)\)$ is not a single fixed number. It depends entirely on whether $x$ is a multiple of 3 and whether $y$ is a multiple of 2 .
3. Conclusion
Since the result changes depending on the specific values of $x$ and $y$ chosen, the correct answer is A. Cannot be determined.
gmatclubot
Re: If $x$ and $y$ are positive integers, and 1 is the greatest common div [#permalink]
20 Feb 2026, 14:00
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