Q&A -Ask Doubts and Get AnswersPrevious Next
Real Numbers Solutions
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Proof: Suppose a is any positive integer and b = 3. By using Euclid’s algorithm we get a = 3q + r , for some integer q 0, and r = 0,1,2 because of 0 r 3.
Now substituting the value of r, we get,
If r = 0, then a = 3q
Similarly, for r = 1 and 2, the value of a is 3q + 1 and 3q + 2 respectively.
Now, we can find the cube of values 3q , 3q + 1 and 3q + 2.
So, we have the following cases.
Case I: When a = 3q
(3q)3 = 27q3 = 9 (3q3) = 9m, if we divide by 9, then the remainder is 0, where m = 3q3.
Case II: When a = 3q + 1
(3q + 1)3
= 27q3 + 27q2 + 9q + 1 [ (x + y)3 = x3 + 3x2y + 3xy2 + y3 ]
= 9 q (3q2 + 3q + 1) + 1
= 9m + 1, if we divide by 9, then the remainder is 1 and the value of m is q(3q2 + 3q + 1).
Case III: When a = 3q + 2
(3q + 2)3
= 27 q3 + 54q2 + 36q + 8 [ (x + y)3 = x3 + 3x2y + 3xy2 + y3 ]
= 9q (3q2 + 6q + 4) + 8
= 9 m + 8, if we divide by 9, then the remainder is 8 and the value of m is q (3q2 + 6q + 4).
Hence, the cube of any positive integer is either in form of 9 m or 9 m + 1 or, 9 m + 8.
Related Questions for Study
What our students and parents say about us!
Saiyam is using EduSaksham online courses from past one year. He is able to complete his tutorials, worksheets, and assignments on his own. Once he is done with worksheets, he can check his answers and if he has done any mistakes, then he can check the correct answers and explanations. They have been given so well that he is able to understand them himself and rarely he asks us for any clarification. He enjoys using the platform and has learned and improved a lot in Mathematics. I would recommend EduSaksham to all parents who want their kids to learn in a structured way and also enjoy the learning process.
Neeraj (Saiyam's Father)
I would like to thank Edusaksham on behalf of my son Aditya who attended " Math Olympiad crash course for 5 weeks". As per Aditya, this course helped him to understand new topics like Roman numerals and decimal places which were not taught in his school till date and had a better understanding of other topics. I personally thank EduSaksham for all the help and support.
Geeta Bansal (Aditya's Mother)
We had a good experience with EduSaksham. Because sheets are available online, Vedant enjoyed doing more and more sheets without any reminders. I could leave him on his own as he was very much motivated to cover the topics. Practice is very important in maths but it becomes difficult for us to prepare sheets, I am glad we found EduSaksham where in a lot of in a lot of sheets are available for practice and the quality of questions are also good. As it's a venture from IITians we could think of giving it a try and I am glad we tried
Shikha Chowdhary (Vedant's Mother)
I have grown up in a family where my father used to give me practice maths paper every day which I need to finish before I go to play. I think that strong foundation of solving new questions every day helped me to clear my IITJEE exam. But as working women, I always had time challenge to create practice Math papers for my daughter. Thanks to EduSaksham, their questions are so logical and conceptual that I don't feel the guilt of not creating practice questions by myself :). There are plenty of worksheet along with the tutorials for each chapter. It has made parents life easier :) And the best part is my daughter loves doing it. Thank you EduSaksham.