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For positive integers f(n), let n return the smallest positive integer k such that 1/k has exactly n digits after the decimal point. How many positive integer divisors does f(2010) have?

 Mar 27, 2018
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f(n)=2n.

 

Proof-

Suppose B is a number with a finite number of digits after the decimal point. here's his decimal representation:

 

0.r1r2r3.....rk0000000000......

 

 where r1,r2,r3.....rk are digits. This means that we can write B as a quotient of 2 positive integers:

p=r1*10k-1+r2*10k-2+......rk-1*101+rk*100, q=10k

B=p/q

 

this means that if we have a rational number C, he has a finite number of digits after the decimal point IF AND ONLY IF C=x/(10n) where n is some natural number (think about it). C has EXACTLY n digits after the decimal point if n is the minimal natural number that satisfies this equation (if C is a number between 0 and 1 this means that x is not divisible by 10).

 

if C's simplified form as a fraction is p1/q1 this means that the only divisors q1 can have are the divisors of 10: meaning we can write qas the product 2a*5b where a and b are natural numbers. if q1=2a*5b this means that the smallest k that will satisfy the equation x/10k=C is k=max(a,b).

 

a fraction of the form 1/k is already simplified. the fraction having a finite number of digits after the decimal point means that k=2a*5b meaning that he has exactly max(a,b) digits after the decimal point. we need to find the minimal k that satisfies k=2a*5b and max(a,b)=n, and that k is k=2n. why? because if a=n this means that the equation max(a,b)=n is already satisfied- we don't need k to be divisible by 5's, and the smallest number that is divisible by 2n is 2n itself.

 

if a a*5 n>=5 n>=2 n

 

if f(n)=2n then f(n) has exactly n positive integer divisors (the divisors are: 21,22,23,.....,2n)

 

so the answer to your question is:

 

f(2010) has 2010 positive divisors

 Mar 29, 2018
edited by Guest  Mar 29, 2018

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