A rectangular field has dimensions $120$ meters and $192$ meters. You want to divide it into equal square plots. The measure of the sides of these squares must be an integer number . In addition, you want to place a post in each corner of plot. Determine the smallest number of plots in which you can divide the land and the number of posts needed. Original wordingUn terreno de forma rectangular de 120 metros por 192 metros se quiere dividir en parcelas cuadradas iguales sin que sobre terreno. La medida de los lados de estos cuadrados debe ser un nu´mero entero. Adem´as se desea colocar un poste en cada esquina de parcela. Determinar el menor nu´mero de parcelas en que se puede dividir el terreno y el nu´mero de postes que se necesitan.

Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

Five persons of different heights stand next to the another on numbered booths to take a picture. From how many ways can be arranged so that people in positions $ 1$ and $3$ are both taller than the person in the position $2$?

It basically says that in a circle, if two chords $ AB$ and $ CD$ intersect at a point $ M$, then $ AM \cdot MB = CM \cdot MD$.

Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

The geometric mean of a set of $m$ non-negative numbers is the $m$-th root of the product of these numbers. For which positive values of $n$, is there a finite set $S_n$ of $n$ positive integers different such that the geometric mean of any subset of $S_n$ is an integer?