18 2 If point \( A \) is located at \( (-3,-1) \) and there are 10 lf points between points \( A \) and \( B \), what could bi the possible coordinates for point \( B \) ?

We are told that point A is (–3, –1) and that there are 10 lattice points (points with integer coordinates) strictly between A and B on the line segment from A to B. (The endpoints A and B are not counted among these 10 points.) A useful fact is that if A = (x₁, y₁) and B = (x₂, y₂) are lattice points, then the number of lattice points strictly between them is given by   gcd(|x₂ – x₁|, |y₂ – y₁|) – 1. In our problem, we need this number to be 10. Thus, if we let Δx = x₂ – (–3) and Δy = y₂ – (–1), we have   gcd(|Δx|, |Δy|) – 1 = 10  ⟹  gcd(|Δx|, |Δy|) = 11. Since 11 is a prime number, this tells us that the differences Δx and Δy must be multiples of 11. Write   Δx = 11m  and  Δy = 11n for some integers m and n. Furthermore, the condition gcd(|Δx|, |Δy|) = 11 implies   11·gcd(|m|, |n|) = 11  ⟹  gcd(|m|, |n|) = 1. Thus, the coordinates of B must be of the form   B = (–3 + 11m, –1 + 11n) where m and n are integers that have no common factors other than ±1 (they are “relatively prime”). Here are a few examples: 1. If m = 1 and n = 0, then   B = (–3 + 11, –1 + 0) = (8, –1). 2. If m = 0 and n = 1, then   B = (–3 + 0, –1 + 11) = (–3, 10). 3. If m = 1 and n = 1, then   B = (–3 + 11, –1 + 11) = (8, 10). Any point B of the form (–3 + 11m, –1 + 11n) with gcd(|m|, |n|) = 1 will work. Thus, one possible answer is: B = (8, –1).