be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
Y by Adventure10, Mango247
For a certain positive integer 
less than 
, the decimal equivalent of 
is 
, a repeating decimal of period 
, and the decimal equivalent of 
is 
, a repeating decimal of period 
. In which interval does lie?
![$\textbf{(A)}\ [1,200] \qquad
\textbf{(B)}\ [201,400] \qquad
\textbf{(C)}\ [401,600] \qquad
\textbf{(D)}\ [601,800] \qquad
\textbf{(E)}\ [801,999] $](//latex.artofproblemsolving.com/5/7/8/578961e8796b09bfacecfbc99a092b7dd78a2905.png)
less than 
, the decimal equivalent of 
is 
, a repeating decimal of period 
, and the decimal equivalent of 
is 
, a repeating decimal of period 
. In which interval does lie?![$\textbf{(A)}\ [1,200] \qquad
\textbf{(B)}\ [201,400] \qquad
\textbf{(C)}\ [401,600] \qquad
\textbf{(D)}\ [601,800] \qquad
\textbf{(E)}\ [801,999] $](http://latex.artofproblemsolving.com/5/7/8/578961e8796b09bfacecfbc99a092b7dd78a2905.png)
and 
.
)
.
fails, so 
,
as the answer.
and 
.
clearly has period 2
should divide 
?
,
,
,
,
Subtracting the first from the second, we have that 
Hence since 
is an integer, we have that 
and 
.
.
.
,
is a positive integer that divides 
and 
.
where 
divisible by 7?
