Home
Class 11
MATHS
Using binomial theorem, prove that 6^n-5...

Using binomial theorem, prove that `6^n-5n` always leaves remainder 1 when divided by 25.

Text Solution

Verified by Experts

`6^(n)-5n =(1+5)^(n) -5n`
`= (1+5n+.^(n)C_(2).5^(2)+.^(n)C_(3)5^(3)+"…..")-5n`
`= 25(.^(n)C_(2)+.^(n)C_(3)+"……")+1`
Hence, `6^(n) - 5n` when divided by 25 leaves 1 as remainder.
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS AND QUADRATIC EQUATIONS

    CENGAGE ENGLISH|Exercise All Questions|886 Videos

Similar Questions

Explore conceptually related problems

Using binomial theorem, prove that 6^n-5n always leaves he remainder 1 when divided by 25.

Using binomial theorem, prove that 8^(n)-7n always leaves remainder 1 when divided by 49.

With the help of bionomial expansion, show that (4^(n)-3n) is always leaves remainder 1, when divided by 9.

Using binomial theorem, evaluate : (99)^5

Using binomial theorem compute : (99)^5

Using binomial theorem, evaluate : (102)^5

Using binomial theorem compute : (102)^6

Using binomial theorem compute : (10. 1)^5

What is the remainder when 13 is divided by 6?

What will be the remainder when 5^97 is divided by 52

CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. Find the remainder when 5^(99) is divided by 13.

    Text Solution

    |

  2. Find the remainder when 7^(103) is divided by 25.

    Text Solution

    |

  3. Using binomial theorem, prove that 6^n-5n always leaves remainder 1...

    Text Solution

    |

  4. If the coefficient of the middle term in the expansion of (1+x)^(2n+2)...

    Text Solution

    |

  5. If the coefficients of three consecutive terms in the expansion of (1+...

    Text Solution

    |

  6. In the coefficients of rth, (r+1)t h ,a n d(r+2)t h terms in the binom...

    Text Solution

    |

  7. Prove that (C0+C1)(C1+C2)(C2+C3)(C3+C4)...........(C(n-1)+Cn) = (C0...

    Text Solution

    |

  8. If a1,a2, a3, a4 be the coefficient of four consecutive terms in the e...

    Text Solution

    |

  9. Find the sum of sum(r=1)^n(r^n Cr)/(^n C(r-1) .

    Text Solution

    |

  10. Find the positive integer just greater than (1+0. 0001)^(10000)dot

    Text Solution

    |

  11. Find (i) the last digit, (ii) the last two digits, and (iii) the last ...

    Text Solution

    |

  12. If 10^m divides the number 101^(100)-1 then, find the greatest value o...

    Text Solution

    |

  13. Using the principle of mathematical induction, prove that (2^(3n)-1) i...

    Text Solution

    |

  14. If x is very large as compare to y , then prove that sqrt(x/(x+y))dot...

    Text Solution

    |

  15. Find the coefficient of x^n in the expansion of (1-9x+20 x^2)^(-1)dot

    Text Solution

    |

  16. Prove that the coefficient of x^r in the expansion of (1-2x)^(-1/2) is...

    Text Solution

    |

  17. Find the sum: 1-1/8+1/8xx3/(16)-(1xx3xx5)/(8xx16xx24)+...

    Text Solution

    |

  18. Show that sqrt3 =1+1/3+(1/3).(3/6)+(1/3)*((3/6)*(5/9)*(7/12)+.......

    Text Solution

    |

  19. Assuming x to be so small that x^2 and higher power of x can be neg...

    Text Solution

    |

  20. Find the sum sumsum(0lt=i < jlt=n-1)j^n Cidot

    Text Solution

    |