A professor returned to his home USA after attending a conference in paris and london. He have some money left in euros and pounds sterling. he want to traded both of them into dollars. he received 117.98 dollars with the exchange rate of 1 euro is 11.1 dollars and 1 pound is 1.69 dollars. how nominal euro and pound that he exchanged? In this diophantine equation i can write it euros is x, pounds is y so $11.1x+1.69y=117.98$ $1110x+169y=11798$ $gcd(1110,169)=1$
$1|11798$ is true then i find $(x',y')$ $1=-44(1110)+289(169)$ $11798=-519112(1110)+3409622(169)$
If we talk about money so $x>=0$ and $y>=0$
$x=-519112+169n$ $x>=0$ $-519112+169n>=0$ $n>=3071.6$ So i find $n={3072,3073,...}$ Then for y $y=3409622-1110n$ $y>=0$
With same method like x i find that $n={...,3070,3071}$
What i got from this is that x and y not both positive. Is that possible?
As per my comment, use euro to dollars rate of $1.11$
Clear the cents to give $111x+169y=11798$
Solve, either with the method shown in your question, or just cheat with http://planetcalc.com/3303/
Either way,
$$x=790466+169k$$
$$y=-519112-111k$$
Clearly, $y>0$, so calculate $-519112/111=-4676.68$
Try $k=-4677$ to give $x=53,y=35$
Now $k=-4678$ and smaller makes $x$ negative.
Also, $k=-4676$ and larger makes $y$ negative, so there is just one answer.
$53$ euro, $35$ pounds