For every pair of non-zero integers $(a, b)$, we denote by $\operatorname{gcd}(a, b)$ the greatest common divisor of $a$ and $b$. The plane is equipped with a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.
- Example. Let $\Delta_{1}$ be the line with equation $y = \frac{5}{4}x - \frac{2}{3}$. a. Show that if $(x, y)$ is a pair of integers then the integer $15x - 12y$ is divisible by 3. b. Does there exist at least one point on the line $\Delta_{1}$ whose coordinates are two integers? Justify.
Generalization: We now consider a line $\Delta$ with equation $(E): y = \frac{m}{n}x - \frac{p}{q}$ where $m, n, p$ and $q$ are non-zero integers such that $\operatorname{gcd}(m, n) = \operatorname{gcd}(p, q) = 1$. Thus, the coefficients of equation $(E)$ are irreducible fractions and we say that $\Delta$ is a rational line. The purpose of the exercise is to determine a necessary and sufficient condition on $m, n, p$ and $q$ for a rational line $\Delta$ to contain at least one point whose coordinates are two integers.
\setcounter{enumi}{1} - We suppose here that the line $\Delta$ contains a point with coordinates $(x_{0}, y_{0})$ where $x_{0}$ and $y_{0}$ are integers. a. By noting that the number $ny_{0} - mx_{0}$ is an integer, prove that $q$ divides the product $np$. b. Deduce that $q$ divides $n$.
- Conversely, we suppose that $q$ divides $n$, and we wish to find a pair $(x_{0}, y_{0})$ of integers such that $y_{0} = \frac{m}{n}x_{0} - \frac{p}{q}$. a. We set $n = qr$, where $r$ is a non-zero integer. Prove that we can find two integers $u$ and $v$ such that $qru - mv = 1$. b. Deduce that there exists a pair $(x_{0}, y_{0})$ of integers such that $y_{0} = \frac{m}{n}x_{0} - \frac{p}{q}$.
- Let $\Delta$ be the line with equation $y = \frac{3}{8}x - \frac{7}{4}$. Does this line have a point whose coordinates are integers? Justify.