Trying to understand what is requested in this task and its solution 
A student wishes to travel to the USA and changes amounts of Euro and
  Swiss Franc in Dollar (in whole numbers!). The exchange rates are \$ 1.35 = € 1.00 and \$
  1.12 = CHF 1.00. If the student receives \$ 66.49 in total, then how much money of each currency has been changed?

So I don't really understand this task. The student has got \$ 66.49. 
This seems problematic "how much money of each currency has been changed?".
Am I asked to convert \$ 66.49 into € and CHF?
If so then: $$\frac{\$66.49}{\$1.35}=€49.25$$
and
$$\frac{\$66.49}{\$1.12}=50.37 \text{ CHF}$$
But wait, it was mentioned "in whole numbers"... So what to do? Or the task is just bad? I think it cannot be as easily done as I tried here, no way : /
Edit: I heard there is a recommendation using extended euclidean algorithm but I don't see the use of it here at all.
 A: Scaling by $\,100\,$ we need to solve $\ 112x + 135 y = 6649\ $ for $\,x,y\in\Bbb N.\,$ Using Gauss's algorithm
$\quad {\rm mod}\ 112\!:\,\ y \equiv \dfrac{6649}{135}\equiv \dfrac{41}{23}\,\overset{\times 5}\equiv\, \dfrac{205}{115}\equiv \dfrac{93}3\equiv 31,\ $ so $\,\  x = \dfrac{6649-135\cdot 31}{112} = 22$

Beware $ $ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.
A: The problem is asking you to find an integer amount of Euros and an integer amount of Swiss Franc which when converted gives you USD 66.49, so you need to solve
$1.35e + 1.12f = 66.49$
where $e$ and $f$ are integers, which, as someone else has already said in the comments, is a diophantine equation. 
Now, I don't know how to properly solve a diophantine equation, but I have found a solution. 
You know that the final digit is 9, and since multiplying 1.35 will only ever get you final digits 0 or 5 and multiplying 1.12 will never get you final digit 9, it must be that $e$ is odd and $f$ will get you a 4. 
So, first I tried to find an odd integer $a$ such that
$1.35 a + 2.24 = n + 0.49$
where n in any integer. I literally got a calculator and attempted values and found $a = 15$ and $n = 22$. So now we have 22.49 and we need 44 still. 
Since 44 is an integer you know you have to multiply 1.12 by a multiple of 5 and 1.35 by an even number. $1.12 \times 5 = 5.6$ and $1.35 \times 4 = 5.4 $ so that 
$1.12 \times 5 + 1.35 \times 4 = 11$
If you multiply that by 4 you have the missing 44 and finally
$ 1.35 \times 15 + 1.12 \times 2 + 4 \times (1.12 \times 5 + 1.35 \times 4) = 31 \times 1.35 + 22 \times 1.12 = 66.49$
Thus,
$e = 31$ 
$f = 22$
