Question: The point of intersection of $4x+5y=26$ and $y=kx+2$ has integral coordinates. What is the number of integral values that $k$ can take?
As per me the answer should be that $k$ can take $3$ integral values which will make coordinates of the point of intersection of the $2$ equations integer.
How I did this is by equating the equations. Then I got $x = \frac{16}{4+5k}$
Now $16$ has $5$ positive factors $(1, 2, 4, 8, 16)$ and $x$ can even take negative values
Then $5k+4$ can have values ranging from $1, 2, 4, 8, 16$ and $-1, -2, -4, -8, -16$
We will see that $k$ has an integral at $5k+4 = 4, -1, -16$ which makes $3$ values for $k$ Is this correct since the answer is given as $1$ which is not true has I got $k = 0$ and $-1$ which satisfies the conditions and the true answer should be $3$?
Plus if this is correct, any shorter way for this?