<s>[INST] $5 \frac {3}{*} \times 3 \frac {*}{2}=19$? One of my friends gave me this apparently easy-looking problem which I do not know how to crack. The problem is to find the values of   "*" where 
$$5 \frac {3}{*} \times 3 \frac {*}{2}=19\text{ ?}$$ 
I can rearrange the problem as $5 \frac {3}{x} \times 3 \frac {y}{2}=19$ and I have to find the values of $x,y.$  Now $5 \frac {3}{x} \times 3 \frac {y}{2}=19 \implies \frac {5x+3}{x} \times \frac {6+y}{2}=19.$ Now, I do not know which way to  go?
Can someone point me in the right direction? Thanks in advance for your time.
EDIT: Here "*"-s are not same. Infact I know the answer but do not know how to get it. Here $x=7,y=1.$ 
 [/INST] On simplification,  $y=\frac{8x-18}{5x+3}$
Now, $$8(5x+3)-5(8x-18)=114$$
So, $5x+3$ must divide $114$ and as $x>0, 5x+3>3$
Now the factors of $114$ are $1,2,3,6,19,38,57,114$
So,  $5x+3$ can be  $6,19,38,57,114$
$6\equiv1\pmod 5,19\equiv4, 38\equiv3,57\equiv2,114\equiv4$
So,  $5x+3$ can be $38\implies x=7,y=1$
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