Home
Class 11
MATHS
If a\ a n d\ b are distinct integers, pr...

If `a\ a n d\ b` are distinct integers, prove that `a^n-b^n` is divisible by `(a-b)` where `n in NN`.

Text Solution

AI Generated Solution

To prove that \( a^n - b^n \) is divisible by \( a - b \) for distinct integers \( a \) and \( b \), where \( n \) is a natural number, we can use the Binomial Theorem. ### Step-by-Step Solution: 1. **Start with the expression**: We need to prove that \( a^n - b^n \) is divisible by \( a - b \). 2. **Factor the expression using the difference of powers**: ...
Promotional Banner

Topper's Solved these Questions

  • ARITHMETIC PROGRESSIONS

    RD SHARMA|Exercise Solved Examples And Exercises|248 Videos

  • BRIEF REVIEW OF CARTESIAN SYSTEM OF RECTANGULAR COORDINATES

    RD SHARMA|Exercise Solved Examples And Exercises|75 Videos

Similar Questions

Explore conceptually related problems

For any positive integer n,prove that n^(3)-n is divisible by 6

For any positive integer n prove that n^(3)-n is divisible by 6

For any positive integer n, prove that n^(3)-n divisible by 6.

If a and b are distinct integers,prove that a-b is a factor of a^(n)-b^(n), wherever n is a positive integer.

If a and b are distinct integers then prove that (a-b) is a factor of (a^(n)-b^(n)) , whenever n is a positive integar.

If n is an even positive integer, then a^(n)+b^(n) is divisible by

If n is an odd positive integer, then a^(n)+b^(n) is divisible by

If and b are distinct integers,prove that a-b is a factor of whenever n is a positive integer.

If a>b and n is a positive integer,then prove that a^(n)-b^(n)>n(ab)^((n-1)/2)(a-b)

Consider the following statements: 1. a^n+b^n is divisible by a+b if n=2k+1, where k is a positive integer. 2. a^n-b^n is divisible by a-b if n=2k , where k is a positive integer. Which of the statements given above is /are correct? (a) 1 only (b) 2 only (c) both 1 and 2 (d) Neither 1 nor 2

RD SHARMA-BINOMIAL THEOREM-Solved Examples And Exercises
  1. Using binomial theorem, prove that 6^n-5n always leaves he remainder 1...

    Text Solution

    |

  2. Which is larger (1 0.1)^(100000)\ or 10,000?

    Text Solution

    |

  3. If a\ a n d\ b are distinct integers, prove that a^n-b^n is divisible ...

    Text Solution

    |

  4. Using binomial theorem, write down the expansion of (2x+3y)^5

    Text Solution

    |

  5. Using binomial theorem, write down the expansions of the following: (1...

    Text Solution

    |

  6. Using binomial theorem, write down the expansions of the following: ...

    Text Solution

    |

  7. Using binomial theorem, write down the expansions of the following: ...

    Text Solution

    |

  8. Using binomial theorem, write down the expansions of the following: (2...

    Text Solution

    |

  9. Using binomial theorem, write down the expansions of the following: \ ...

    Text Solution

    |

  10. Using binomial theorem, write down the expansions of the following: \ ...

    Text Solution

    |

  11. Using binomial theorem, write down the expansions of the following: \ ...

    Text Solution

    |

  12. Using binomial theorem, write down the expansions of the following: \ ...

    Text Solution

    |

  13. Using binomial theorem, write down the expansions of the following:\ (...

    Text Solution

    |

  14. Evaluate the following: (sqrt(x+1)+sqrt(x-1))^6+(sqrt(x+1)-sqrt(x-1))^...

    Text Solution

    |

  15. Evaluate the following: \ (3+sqrt(2))^5-(3-sqrt(2))^5

    Text Solution

    |

  16. Evaluate the following: \ (1+2sqrt(x))^5+(1-2sqrt(x))^5

    Text Solution

    |

  17. Evaluate the following: \ (sqrt(3)+1)^5-(sqrt(3)-1)^5

    Text Solution

    |

  18. Evaluate the following: \ (sqrt(3)+sqrt(2))^6-(sqrt(3)-sqrt(2))^6

    Text Solution

    |

  19. Evaluate the following: \ {a^2+sqrt(a^2-1)}^4+{a^2-sqrt(a^2-1)}^4

    Text Solution

    |

  20. Evaluate the following: \ (x+sqrt(x^2-1))^6+(x-sqrt(x^2-1))^6

    Text Solution

    |