CEMC Fermat, 2022

Suppose that $k$ is a fixed,
but unknown, positive integer.

Suppose also that the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at the point with
positive integer coordinates $(x,y)$.

Since $9x+4y=600$ and $kx-4y=24$, adding these equations, we get
$9x + kx = 624$ and so $(9+k)x=624$.

Since $x$ and $y$ are to be positive integers and $\text{k gt;0}$, then $9+k$ and $x$ are a positive divisor pair of 624 with
$\text{9+k gt;9}$.

Now $624 = 6 \cdot 104 = 6 \cdot 8 \cdot 13 = 2^4 3^1 13^1$, and so the positive divisors of $624$ are $$1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624$$ We also want the value of $y$ to be a positive integer.

Since the point $(x,y)$ lies on the
line with equation $9x + 4y = 600$,
then $4y = 600 - 9x$ which gives
$y = 150 - \frac{9}{4}x$, which is an
integer exactly when $x$ is a
multiple of 4.

Therefore, we want $x$ to be a
positive divisor of 624 which is a multiple of 4.

Thus, the possible values of $x$ are
$4, 8, 12, 16, 24, 48, 52, 104, 156, 208, 312, 624$.

The corresponding values of $9+k$ are
$156, 78, 52, 39, 26, 13, 12, 6, 4, 3, 2, 1$.

Since $\text{9+k gt;9}$, we eliminate $6, 4, 3, 2, 1$ from this list.

Thus, the possible values of $9+k$
are $156, 78, 52, 39, 26, 13, 12$.

The corresponding values of $k$ are
$147, 69, 43, 30, 17, 4, 3$.

These correspond to the following values of $x$: $4, 8, 12, 16, 24, 48, 52$.

Using $y = 150 - \frac{9}{4}x$, these
give the following values of $y$:
$141, 132, 123, 114, 96, 42, 33$.
These are indeed all positive.

This means that there are 7 values of $k$ with the required properties.