Review question

# How many grid-points can be inside this circle? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7626

## Question

Suppose that $a$, $b$, $c$ are integers such that $a \sqrt{2} + b = c \sqrt{3}.$

1. By squaring both sides of the equation, show that $a = b = c = 0$.

[You may assume that $\sqrt{2}, \sqrt{3}$ and $\sqrt{2/3}$ are all irrational numbers. An irrational number is one which cannot be written in the form $p/q$ where $p$ and $q$ are integers.]

2. Suppose now that $m$, $n$, $M$, $N$ are integers such that the distance from the point $(m,n)$ to $(\sqrt{2},\sqrt{3})$ equals the distance from $(M,N)$ to $(\sqrt{2},\sqrt{3})$.

Show that $m=M$ and $n=N$.

Given real numbers $a$, $b$ and a positive number $r$, let $N(a,b,r)$ be the number of integer pairs $x$, $y$ such that the distance between the points $(x,y)$ and $(a,b)$ is less than or equal to $r$. For example, we see that $N(1.2,0,1.5)=7$ in the diagram below.

1. Explain why $N(0.5,0.5,r)$ is a multiple of $4$ for any value of $r$.

2. Let $k$ be any positive integer. Explain why there is a positive number $r$ such that $N(\sqrt{2},\sqrt{3},r)=k$.