Béla has three circular templates: these are suitable for drawing circles with areas of 6, 15, and $83 \mathrm{~cm}^{2}$ respectively. Béla wants to draw some circles, the total area of which should be $220 \mathrm{~cm}^{2}$. How many of each circle should he draw?
Solution. The largest circle with an area of $83 \mathrm{~cm}^{2}$ can have at most two drawn from it. If we do not draw any from this, then the total area of the circles with areas of $6 \mathrm{~cm}^{2}$ and $15 \mathrm{~cm}^{2}$ will be divisible by $3$, so it cannot be $220 \mathrm{~cm}^{2}$. If we draw one from the largest circle, then the remaining $137 \mathrm{~cm}^{2}$ area also cannot be produced from the areas of the other two circles, because 137 is not divisible by 3. Therefore, we must definitely draw two from the largest circle. In this case, for the remaining $54 \mathrm{~cm}^{2}$ area, we need 9 circles with an area of $6 \mathrm{~cm}^{2}$, or 4 circles with an area of $6 \mathrm{~cm}^{2}$ and 2 circles with an area of $15 \mathrm{~cm}^{2}$.