Task B-4.8. Tomislav, Filip, and Nikola came to a large shopping center with their girlfriends Ana, Marija, and Iva to go shopping. Each of the six paid as many kuna for each item as the number of items they bought, and each young man spent 63 kuna more than his girlfriend. If Tomislav bought 23 more items than Marija, and Nikola bought 11 more items than Iva, determine all the pairs.
Solution. Let $x$ be the number of items the young man bought, and $y$ be the number of items his girlfriend bought.

For each couple, the equation is $x^{2}-y^{2}=63$.

(2 points)

From $(x-y) \cdot(x+y)=63$, we easily find three solutions:

$$
x=32, y=31 \quad \text { or } \quad x=12, y=9 \quad \text { or } \quad x=8, y=1
$$

Since Tomislav bought 23 more items than Marija, we conclude that Tomislav bought 32, and Marija 9 items.

In the same way, Nikola bought 12 items, and Iva 1 item.

In the end, we conclude that Filip bought 8 items, and Ana 31 items.