Task 5. Elena and Desanka together have six 50 denar coins and four 100 denar banknotes. In how many ways can they equally divide the money.
Solution. Elena and Desanka together have $6 \cdot 50 + 4 \cdot 100 = 700$ denars, so each of them should receive $700 : 2 = 350$ denars. Let $x$ be the number of 50 denar coins, and $y$ be the number of 100 denar banknotes that Elena received. Then $50x + 100y = 350$, or $x + 2y = 7$, where $x \in \{0, 1, 2, 3, 4, 5, 6\}$ and $y \in \{0, 1, 2, 3, 4\}$. From $x + 2y = 7$, we conclude that $x$ is an odd number, i.e., $x \in \{1, 3, 5\}$.

If $x = 1$, then $y = (7 - 1) : 2 = 3$, which means that Elena received 1 coin of 50 denars and 3 banknotes of 100 denars. If $x = 3$, then $y = (7 - 3) : 2 = 2$, which means that Elena received 3 coins of 50 denars and 2 banknotes of 100 denars. If $x = 5$, then $y = (7 - 5) : 2 = 1$, which means that Elena received 5 coins of 50 denars and 1 banknote of 100 denars.

Finally, Elena and Desanka can divide the money in the three described ways.

## VII section