text: # Difference between revisions of "2007 AIME I Problems/Problem 1"
  
  ## Problem
  
  How many positive perfect squares less than $10^6$ are multiples of $24$?
  
  ## Solution
  
  The prime factorization of $24$ is $2^3\cdot3$. Thus, each square must have at least $3$ factors of $2$ and $1$ factor of $3$ and its square root must have $2$ factors of $2$ and $1$ factor of $3$. This means that each square is in the form $(12c)^2$, where $12 c$ is a positive integer less than $\sqrt{10^6}$. There are $\left\lfloor \frac{1000}{12}\right\rfloor = \boxed{083}$ solutions.
  
  ## Solution 2
  
  After some work you will eventually find its just $\left\lfloor \frac{1000}{12}\right\rfloor = \boxed{083}$ solutions.
explanation: The document is a revision of a problem and its solutions from the 2007 AIME I math competition. It contains a math problem and two different solutions to the problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Difference between revisions of "2007 AIME I Problems/Problem 1"
  
  ## Problem
  
  How many positive perfect squares less than $10^6$ are multiples of $24$?
  
  ## Solution
  
  The prime factorization of $24$ is $2^3\cdot3$. Thus, each square must have at least $3$ factors of $2$ and $1$ factor of $3$ and its square root must have $2$ factors of $2$ and $1$ factor of $3$. This means that each square is in the form $(12c)^2$, where $12 c$ is a positive integer less than $\sqrt{10^6}$. There are $\left\lfloor \frac{1000}{12}\right\rfloor = \boxed{083}$ solutions.
  
  ## Solution 2
  
  After some work you will eventually find its just $\left\lfloor \frac{1000}{12}\right\rfloor = \boxed{083}$ solutions.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a revision of a problem and its solutions from the 2007 AIME I math competition. It contains a math problem and two different solutions to the problem.
  
  Label: question-and-answer