Buying exactly 100 animals for 1000 dollars

At a market, a farmer pays exactly $1000 for exactly $100$ animals, buying at least one cow, one sheep and one chicken. Cows cost $50, sheep $10 and chickens 50 cents each. If he bought $x_1$ cows, $x_2$ sheep and $x_3$ chickens, what is $x_1x_2x_3$?

1520. From $x_1+x_2+x_3=100$ and $50x_1+10x_2+0.5x_3=1000$, eliminating $x_3$ gives $99x_1+19x_2=1900$. So $19\mid x_1$; $x_1=19$ forces $x_2=1$ (and $x_1=0$ is excluded), hence $x_3=80$ and the product is $19\cdot1\cdot80=1520$.