text: 1. ## Langrange theorem problem....
  
  Find [ Z_40 : <[12], [20]> ]?
  
  I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.
  
  Thanks for the help!
  
  2. ## Re: Langrange theorem problem....
  
  Originally Posted by vedicmath
  Find [ Z_40 : <[12], [20]> ]?
  
  I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.
  
  Thanks for the help!
  So, it is "well known" that any subgroup of a cyclic group is cyclic. As you have pointed out, gcd(12, 20)=4. This means that the subgroup generated by 12 and 20 is equal to the subgroup generated by 4, $\displaystyle \langle 12, 20\rangle=\langle 4\rangle$ (this should be in your notes, and is the crux of the argument - if you do not know it, learn it!). So, what is the index of $\displaystyle \langle 4\rangle$ in $\displaystyle \mathbb{Z}_4$? Divide 40 by this order to get your answer (why?).
explanation: The document contains a math question and an answer to that question. The question is about finding the index of a subgroup in a cyclic group, and the answer provides a detailed explanation of how to proceed using the greatest common divisor (gcd) and properties of cyclic groups.
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distilabel_metadata.content: 1. ## Langrange theorem problem....
  
  Find [ Z_40 : <[12], [20]> ]?
  
  I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.
  
  Thanks for the help!
  
  2. ## Re: Langrange theorem problem....
  
  Originally Posted by vedicmath
  Find [ Z_40 : <[12], [20]> ]?
  
  I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.
  
  Thanks for the help!
  So, it is "well known" that any subgroup of a cyclic group is cyclic. As you have pointed out, gcd(12, 20)=4. This means that the subgroup generated by 12 and 20 is equal to the subgroup generated by 4, $\displaystyle \langle 12, 20\rangle=\langle 4\rangle$ (this should be in your notes, and is the crux of the argument - if you do not know it, learn it!). So, what is the index of $\displaystyle \langle 4\rangle$ in $\displaystyle \mathbb{Z}_4$? Divide 40 by this order to get your answer (why?).
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and an answer to that question. The question is about finding the index of a subgroup in a cyclic group, and the answer provides a detailed explanation of how to proceed using the greatest common divisor (gcd) and properties of cyclic groups.
  
  Label: question-and-answer