The question is , if f(n)= 1+10+10^2 + + 10^n , where n is

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Discussion Overview

The discussion centers on the function f(n) defined as the sum of a geometric series: f(n) = 1 + 10 + 10^2 + ... + 10^n, where n is an integer. Participants are exploring how to determine the least integer n such that f(n) is divisible by 17. The scope includes mathematical reasoning and problem-solving approaches.

Discussion Character

  • Mathematical reasoning, Homework-related, Exploratory

Main Points Raised

  • One participant expresses uncertainty about how to find the least n such that f(n) is divisible by 17.
  • Another participant suggests calculating the remainders of powers of 10 when divided by 17, proposing that this could aid in finding a solution.
  • A different participant mentions that recognizing a relationship involving 9 times the sum plus 1 being a power of 10 could be relevant, hinting at the applicability of Fermat's Little Theorem.
  • One participant proposes a "brute force" method to check the divisibility of f(n) for successive values of n, suggesting the use of computational tools or programming for this task.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on a single method for solving the problem, with multiple approaches and suggestions being presented.

Contextual Notes

Some assumptions regarding the properties of modular arithmetic and the applicability of Fermat's Little Theorem are not fully explored or established in the discussion.

sigh1342
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the question is , if f(n)= 1+10+10^2 +... + 10^n , where n is integer.
find the least n s.t. f(n) is divisible by 17 , I have no idea about it.
 
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What is the remainder if you divide...
1 by 17?
10 by 17?
100 by 17?
...
a+b by 17, if you know it for a and b?

That should help.
 


sigh1342 said:
the question is , if f(n)= 1+10+10^2 +... + 10^n , where n is integer.
find the least n s.t. f(n) is divisible by 17 , I have no idea about it.

It should also help to recognize that 9 times the sum plus 1 is a power of 10. Thus Fermats Little theorem re primes P dividing A^(P-1) - 1 may apply.
 


You can also find n by "brute force", i.e. check if f(1), f(2), f(3), ... is divisible by 17. Use http://www.wolframalpha.com/ or write a small program in your programming language of choice.
 
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