Lattice points on parametric equations The definition of a lattice point is a point that has integer coordinates. 
Find the number of lattice points  $(x,y)$, where $-100\le x\le 100$ and $-100\le y\le 100$ are on the graph of the parametric equations \begin{align*} x&=30-40\cos t, y=-50 + 30\cos t. \end{align*}
How would I begin? There could be infinite possibilities for $t$, so I am not sure how to begin this problem.

You can basically ignore the cosine element - just think of it as a variable that can take any value between $-1$ and $1$ inclusive. The equations then define a line section between $(70,-80)$ and $(-10, -20)$:
$$\left .\begin{align}x&=30-40v\\ y&=-50 + 30v \end{align}\right \rbrace v\in [-1,1]$$
Then you just need to find values of $v$ in that range that make $40v$ and $30v$ both integers. 
Clearly for $30v$ to be an integer, you need $v=\frac{k}{30}$, which gives 61 possibilities in range. Simultaneously you need $v=\frac{m}{40}$. This means that $30m=40k \implies 3m=4k$. Since $3$ and $4$ are co-prime, this will only happen when $3 \mid k$ and $4\mid m$, which means that the only solutions are for $v=\frac{n}{10}, n=\{-10,\ldots 10\}$
This gives a total of $21$ solutions.
