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

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SUMMARY

The function f(n) = 1 + 10 + 10^2 + ... + 10^n is analyzed to determine the smallest integer n such that f(n) is divisible by 17. The discussion highlights the utility of Fermat's Little Theorem in relation to prime numbers and suggests a brute force method to find n by checking the divisibility of f(n) for increasing values of n. Additionally, it emphasizes the importance of calculating the remainders of powers of 10 when divided by 17 to aid in the solution.

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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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