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Questions on Number Theory - LCM HCF

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  • Rajesh Balasubramanian
    Hi all, Given below are a few questions on LCM and HCF. As ever, they are also available on our blog -
    Message 1 of 4 , Nov 9, 2010
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      Hi all,
       
      Given below are a few questions on LCM and HCF. As ever, they are also available on our blog - http://iimcat.blogspot.com/2010/11/number-theory-questions-lcm-hcf.html
       
      1. How many pairs of positive integers x,y exist such that HCF of, x,y = 35 and sum of x and y = 1085?

      2. How many pairs of positive integers x,y exist such that HCF(x,y) + LCM (x,y) = 91?

      3. Sum of two numbers x,y = 1050. What is the maximum value of the HCF between x and y?

      4. Sum of two even numbers a and b = 240. If HCF (a,b) is greater than a and b, what is the minimum value of HCF (a,b)?
       
      Happy cracking.
       
      Cheers,
      Rajesh
      99626 48484
      New CAT2011 batches starting @ Chennai
      Nov 13th @ Velachery, Nov 20th @ Mylapore
       
    • kamal lohia
      1. Phi(31) = 30. 2. 8 + 4 + 4 = 16. 3. 1050/2 = 525. 4. HCF(a, b) cannot be greater than a and b. Please confirm the answers. I am sure that there can be some
      Message 2 of 4 , Nov 9, 2010
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        1. Phi(31) = 30.
        2. 8 + 4 + 4 = 16.
        3. 1050/2 = 525.
        4. HCF(a, b) cannot be greater than a and b.

        Please confirm the answers. I am sure that there can be some silly error which i a reluctant to recheck.



        From: Rajesh Balasubramanian <rajesh@...>
        To: ascent4cat@yahoogroups.com
        Sent: Wed, November 10, 2010 9:24:37 AM
        Subject: [2IIM CAT Prep] Questions on Number Theory - LCM HCF

         

        Hi all,
         
        Given below are a few questions on LCM and HCF. As ever, they are also available on our blog - http://iimcat.blogspot.com/2010/11/number-theory-questions-lcm-hcf.html
         
        1. How many pairs of positive integers x,y exist such that HCF of, x,y = 35 and sum of x and y = 1085?

        2. How many pairs of positive integers x,y exist such that HCF(x,y) + LCM (x,y) = 91?

        3. Sum of two numbers x,y = 1050. What is the maximum value of the HCF between x and y?

        4. Sum of two even numbers a and b = 240. If HCF (a,b) is greater than a and b, what is the minimum value of HCF (a,b)?
         
        Happy cracking.
         
        Cheers,
        Rajesh
        99626 48484
        New CAT2011 batches starting @ Chennai
        Nov 13th @ Velachery, Nov 20th @ Mylapore
         

      • divakar narayan
        Ans1: If HCF is 35 then the numbers should be in the form 35k, where k is a prime number. Since it is given x+y=1085, we can write 35(a+b)=1085or a+b = 31. Now
        Message 3 of 4 , Nov 10, 2010
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          Ans1: If HCF is 35 then the numbers should be in the form 35k, where k is a prime number. Since it is given x+y=1085, we can write 35(a+b)=1085
          or a+b = 31. Now only one pair of prime number satisfy this condition(2,29)..hence answere is one pair.(though my gut feeling tells me i am overlooking something so please correct me if i am wrong)
          ANS2:- 16 pairs
          Ans3 :- 525
          Ans4:...I dont know whether it was to be solved or not but HCF cannot be greater than the numbers..please check and send us the question back.


          Opinion is like an AssHole, everybody has one!
           
          Divakar Narayan Singh


          --- On Wed, 10/11/10, Rajesh Balasubramanian <rajesh@...> wrote:

          From: Rajesh Balasubramanian <rajesh@...>
          Subject: [2IIM CAT Prep] Questions on Number Theory - LCM HCF
          To: ascent4cat@yahoogroups.com
          Date: Wednesday, 10 November, 2010, 9:24 AM

           

          Hi all,
           
          Given below are a few questions on LCM and HCF. As ever, they are also available on our blog - http://iimcat.blogspot.com/2010/11/number-theory-questions-lcm-hcf.html
           
          1. How many pairs of positive integers x,y exist such that HCF of, x,y = 35 and sum of x and y = 1085?

          2. How many pairs of positive integers x,y exist such that HCF(x,y) + LCM (x,y) = 91?

          3. Sum of two numbers x,y = 1050. What is the maximum value of the HCF between x and y?

          4. Sum of two even numbers a and b = 240. If HCF (a,b) is greater than a and b, what is the minimum value of HCF (a,b)?
           
          Happy cracking.
           
          Cheers,
          Rajesh
          99626 48484
          New CAT2011 batches starting @ Chennai
          Nov 13th @ Velachery, Nov 20th @ Mylapore
           

        • kamal lohia
          Hi Divakar a + b = 31 such that HCF(a, b) = 1. Now it is not compulsory that a and b are prime. eg. (a, b) can be (1, 30), (2, 29), (3, 28), .....(30, 1) i.e.
          Message 4 of 4 , Nov 10, 2010
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            Hi Divakar
            a + b = 31 such that HCF(a, b) = 1. Now it is not compulsory that a and b are prime. eg. (a, b) can be (1, 30), (2, 29), (3, 28), .....(30, 1) i.e. total 30 pairs.
            Now observe closely that HCF of all these pairs is 1 and their sum is 31.
            Hope it is clear.




            From: divakar narayan <divakar_narayan@...>
            To: ascent4cat@yahoogroups.com
            Sent: Wed, November 10, 2010 7:59:52 PM
            Subject: Re: [2IIM CAT Prep] Questions on Number Theory - LCM HCF

             

            Ans1: If HCF is 35 then the numbers should be in the form 35k, where k is a prime number. Since it is given x+y=1085, we can write 35(a+b)=1085
            or a+b = 31. Now only one pair of prime number satisfy this condition(2,29)..hence answere is one pair.(though my gut feeling tells me i am overlooking something so please correct me if i am wrong)
            ANS2:- 16 pairs
            Ans3 :- 525
            Ans4:...I dont know whether it was to be solved or not but HCF cannot be greater than the numbers..please check and send us the question back.


            Opinion is like an AssHole, everybody has one!
             
            Divakar Narayan Singh


            --- On Wed, 10/11/10, Rajesh Balasubramanian <rajesh@...> wrote:

            From: Rajesh Balasubramanian <rajesh@...>
            Subject: [2IIM CAT Prep] Questions on Number Theory - LCM HCF
            To: ascent4cat@yahoogroups.com
            Date: Wednesday, 10 November, 2010, 9:24 AM

             

            Hi all,
             
            Given below are a few questions on LCM and HCF. As ever, they are also available on our blog - http://iimcat.blogspot.com/2010/11/number-theory-questions-lcm-hcf.html
             
            1. How many pairs of positive integers x,y exist such that HCF of, x,y = 35 and sum of x and y = 1085?

            2. How many pairs of positive integers x,y exist such that HCF(x,y) + LCM (x,y) = 91?

            3. Sum of two numbers x,y = 1050. What is the maximum value of the HCF between x and y?

            4. Sum of two even numbers a and b = 240. If HCF (a,b) is greater than a and b, what is the minimum value of HCF (a,b)?
             
            Happy cracking.
             
            Cheers,
            Rajesh
            99626 48484
            New CAT2011 batches starting @ Chennai
            Nov 13th @ Velachery, Nov 20th @ Mylapore
             


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