One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write:  
\[ \gcd\left(\frac{a}{b},\frac{c}{d}\right) = \frac{\gcd(ad,bc)}{bd}. \]  

For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that:  
\[ \gcd(m,n)=\frac{1}{K}. \]  

What is $f(2017)-f(2016)$?

Let's think step by step and output the final answer within \boxed{}.
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