text: How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
  
  # Cyclic Group problem in NBHM M.Sc. 2013
  
  This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students.
  
  Which of the following statements are true?
  
  1. Every group of order 11 is cyclic.
  2. Every group of order 111 is cyclic.
  3. Every group of order 1111 is cyclic.
  
  Discussion:
  
  1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange's Theorem) that a group of prime order is necessarily cyclic.
  2. Every group of order 111 is cyclic. This is true. $111 = 3 \times 37$. This is a favorite problem for any college level test maker. Order of this group is of the form $p \times q$ where p and q are primes. This group is isomorphic to $Z_p \times Z_q$ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow's Theorem.
  3. Every group of order 1111 is cyclic. This is true. $1111 = 11 \times 101$ and using the argument of the previous problem we conclude that it is cyclic.
  
  Critical Ideas: Lagrange's Theorem, Sylow's Theorem, Classification of finite abelian groups
  
  ### Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com
  
  INMO 2012 Problems
  
  INMO 2018 Problem 6
explanation: The text extract contains a math problem and its solution. Specifically, it discusses a problem from the National Board for Higher Mathematics (NBHM) 2013 exam about cyclic groups. The problem presents three statements about groups of specific orders and asks which of these statements are true. The extract then provides a detailed discussion and justification for each statement, using concepts such as Lagrange's Theorem and Sylow's Theorem.
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: How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
  
  # Cyclic Group problem in NBHM M.Sc. 2013
  
  This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students.
  
  Which of the following statements are true?
  
  1. Every group of order 11 is cyclic.
  2. Every group of order 111 is cyclic.
  3. Every group of order 1111 is cyclic.
  
  Discussion:
  
  1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange's Theorem) that a group of prime order is necessarily cyclic.
  2. Every group of order 111 is cyclic. This is true. $111 = 3 \times 37$. This is a favorite problem for any college level test maker. Order of this group is of the form $p \times q$ where p and q are primes. This group is isomorphic to $Z_p \times Z_q$ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow's Theorem.
  3. Every group of order 1111 is cyclic. This is true. $1111 = 11 \times 101$ and using the argument of the previous problem we conclude that it is cyclic.
  
  Critical Ideas: Lagrange's Theorem, Sylow's Theorem, Classification of finite abelian groups
  
  ### Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com
  
  INMO 2012 Problems
  
  INMO 2018 Problem 6
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math problem and its solution. Specifically, it discusses a problem from the National Board for Higher Mathematics (NBHM) 2013 exam about cyclic groups. The problem presents three statements about groups of specific orders and asks which of these statements are true. The extract then provides a detailed discussion and justification for each statement, using concepts such as Lagrange's Theorem and Sylow's Theorem.
  
  Label: question-and-answer