Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Sequence and greatest integer.
Let S be the sum of all numbers of the form \(\frac{a}{b}\),where a and b are relatively prime positive divisors of 1000, find greatest integer that does not exceed \(\frac{S}{10}\).
Equation
Algebra
Integers
Answer: is 248.
AIME I, 2000, Question 11
Elementary Number Theory by Sierpinsky
We have 1000=(2)(2)(2)(5)(5)(5) and \(\frac{a}{b}=2^{x}5^{y} where -3 \leq x,y \leq 3\)
sum of all numbers of form \(\frac{a}{b}\) such that a and b are relatively prime positive divisors of 1000
=\((2^{-3}+2^{-2}+2^{-1}+2^{0}+2^{1}+2^{2}+2^{3})(5^{-3}+5^{-2}+5^{-1}+5^{0}+5^{1}+5^{2}+5^{3})\)
\(\Rightarrow S= \frac{(2^{-3})(2^{7}-1)}{2-1} \times\) \(\frac{(5^{-3})(5^{7}-1)}{5-1}\)
=2480 + \(\frac{437}{1000}\)
\(\Rightarrow [\frac{s}{10}]\)=248.
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