How many pairs of positive integers $(a,b)$ are there such that $\gcd(a,b)=1$ and \[ \frac{a}{b}+\frac{14b}{9a}
\]is an integer?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
How many pairs of positive integers $(a,b)$ are there such that $\gcd(a,b)=1$ and \[ \frac{a}{b}+\frac{14b}{9a}
\]is an integer?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
4
How many pairs of positive integers $(a,b)$ are there such that $\gcd(a,b)=1$ and \[ \frac{a}{b}+\frac{14b}{9a}
\]is an integer?
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