Class 10 Exam  >  Class 10 Questions  >  Show that a number and its cube leaves the sa... Start Learning for Free
Show that a number and its cube leaves the same remainder when divided by 6.?
Verified Answer
Show that a number and its cube leaves the same remainder when divided...
Suppose a number a is divided by 6 gives a quotient q and remainder r, then we can write,

a=6q+r where  r is an integer such that  


so r takes the values 0,1,2,3,4 and 5.

we see that  cube of each of these numbers leaves a remainder equal to the number itself, when divided by 6.

So,


We see that each of the terms except the last one is a multiple of 6.

In the last term we'll get some quotient say t and a remainder equal to r( as mentioned above)
This question is part of UPSC exam. View all Class 10 courses
Most Upvoted Answer
Show that a number and its cube leaves the same remainder when divided...
Introduction:
To show that a number and its cube leave the same remainder when divided by 6, we need to prove that for any integer 'n', the remainder when 'n' is divided by 6 is the same as the remainder when 'n^3' is divided by 6.

Proof:
Let's consider an integer 'n' and its cube 'n^3'.

Case 1: n is divisible by 6
If 'n' is divisible by 6, then 'n' can be expressed as '6k' for some integer 'k'.
Taking the cube of 'n', we have:
n^3 = (6k)^3 = 6^3 * k^3 = 216k^3

In this case, both 'n' and 'n^3' are divisible by 6, and hence they leave the remainder 0 when divided by 6.

Case 2: n is not divisible by 6
If 'n' is not divisible by 6, then 'n' can be expressed as '6k + r' for some integer 'k' and a remainder 'r' where 0 < r="" />< />
Taking the cube of 'n', we have:
n^3 = (6k + r)^3 = (6k)^3 + 3(6k)^2r + 3(6k)(r^2) + r^3

Expanding the above expression, we have:
n^3 = 216k^3 + 3(36k^2)r + 3(6k)(r^2) + r^3

When we divide 'n^3' by 6, all the terms except 'r^3' are divisible by 6. Hence, the remainder when 'n^3' is divided by 6 is the same as the remainder 'r^3' when 'n' is divided by 6.

Conclusion:
From the above analysis, we can see that for any integer 'n', both 'n' and 'n^3' leave the same remainder when divided by 6. This can be proven using two cases: when 'n' is divisible by 6 and when 'n' is not divisible by 6.
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
Explore Courses for Class 10 exam

Top Courses for Class 10

Show that a number and its cube leaves the same remainder when divided by 6.?
Question Description
Show that a number and its cube leaves the same remainder when divided by 6.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Show that a number and its cube leaves the same remainder when divided by 6.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that a number and its cube leaves the same remainder when divided by 6.?.
Solutions for Show that a number and its cube leaves the same remainder when divided by 6.? in English & in Hindi are available as part of our courses for Class 10. Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of Show that a number and its cube leaves the same remainder when divided by 6.? defined & explained in the simplest way possible. Besides giving the explanation of Show that a number and its cube leaves the same remainder when divided by 6.?, a detailed solution for Show that a number and its cube leaves the same remainder when divided by 6.? has been provided alongside types of Show that a number and its cube leaves the same remainder when divided by 6.? theory, EduRev gives you an ample number of questions to practice Show that a number and its cube leaves the same remainder when divided by 6.? tests, examples and also practice Class 10 tests.
Explore Courses for Class 10 exam

Top Courses for Class 10

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev