Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • If a and b are distinct integers, prove that a - b is a factor of a^n - b^n , whenever n is a positive integer. [Hint write a ^n = (a - b + b) ^n and expand]

Q4.    If a and b are distinct integers, prove thata - b is a factor of a^n - b^n , whenever n is a positive integer.
[Hint: write a^n = (a - b + b)^n and expand]

Answers (1)

best_answer

we need to prove, 

a^n-b^n=k(a-b)  where k is some natural number.

Now let's add and subtract b from a so that we can prove the above result,

a=a-b+b

a^n=(a-b+b)^n=[(a-b)+b]^n

=^nC_0(a-b)^n+^nC_1(a-b)^{n-1}b+........^nC_nb^n

=(a-b)^n+^nC_1(a-b)^{n-1}b+........^nC_{n-1}(a-b)b^{n-1}+b^n\Rightarrow a^n-b^n=(a-b)[(a-b)^{n-1}+^nC_2(a-b)^{n-2}+........+^nC_{n-1}b^{n-1}]

\Rightarrow a^n-b^n=k(a-b)

Hence,a - b is a factor of a^n - b^n.

Posted by

Pankaj Sanodiya

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 12 board exams 2025 to have 50% competency-based questions; no change in 10th
anam-khan

CBSE Class 12 board exams 2024 over; expected result date, trends of last 10 years
anam-khan

CBSE Class 12 accountancy board exam analysis 2024 out; paper rated easy, balanced

Latest Question