A lattice point is a point $(x, y)$ where both $x$ and $y$ are integers For how many different integer values of $k$ will the two lines $k x-5 y+7=0$ and $k^{2} x-5 y+1=0$ intersect at a lattice point?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Subtracting the first equation from the second,

$$
\begin{aligned}
k^{2} x-k x-6 & =0 \\
\left(k^{2}-k\right) x & =6 \\
{[k(k-1)] x } & =6
\end{aligned}
$$

Since we want both $k$ and $x$ to be integers, then $k(k-1)$ is a factor of 6 , ie. is equal to one of $\pm 1, \pm 2, \pm 3, \pm 6$. Now $k(k-1)$ is the product of two consecutive integers, so on this basis we can eliminate six of these eight possibilities to obtain

$$
k(k-1)=2 \quad \text { or } \quad k(k-1)=6
$$

which yields

$$
k^{2}-k-2=0 \quad \text { or } \quad k^{2}-k-6=0
$$

and so $k=2,-1,3,-2$.

We now make a table of values of $k, x$ and $y$ to check when $y$ is also an integer. (We note that from the first equation, $y=\frac{1}{5}(k x+7)$.)

| $k$ | $x$ | $y$ |
| :---: | :---: | :---: |
| 2 | 3 | $\frac{13}{5}$ |
| -1 | 3 | $\frac{4}{5}$ |
| 3 | 1 | 2 |
| -2 | 1 | 1 |

So there are two values of $k$ for which the lines intersect at a lattice point.

ANSWER: (B)