The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.

[i]Proposed by Christopher Cheng[/i]

[hide=Solution][i]Solution. [/i] $\boxed{6}$
Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]
1. **Intersection of Lines**:
   We start by finding the intersection of the lines $\ell_1$ and $\ell_2$. The equations are:
   \[
   24x - 7y = 319
   \]
   \[
   12x - 5y = 125
   \]
   To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we will use elimination.

2. **Elimination Method**:
   Multiply the second equation by 2 to align the coefficients of $x$:
   \[
   24x - 10y = 250
   \]
   Now subtract the second equation from the first:
   \[
   (24x - 7y) - (24x - 10y) = 319 - 250
   \]
   Simplifying, we get:
   \[
   3y = 69 \implies y = 23
   \]
   Substitute $y = 23$ back into the first equation:
   \[
   24x - 7(23) = 319 \implies 24x - 161 = 319 \implies 24x = 480 \implies x = 20
   \]
   Therefore, the intersection point is $(20, 23)$.

3. **Distance from $(20, 23)$**:
   We need to find lattice points on $\ell_1$ and $\ell_2$ that are an integer distance $n$ from $(20, 23)$. The distance formula is:
   \[
   d = \sqrt{(x - 20)^2 + (y - 23)^2}
   \]
   For $d$ to be an integer, $(x - 20)^2 + (y - 23)^2$ must be a perfect square, say $k^2$. Thus:
   \[
   (x - 20)^2 + (y - 23)^2 = n^2
   \]

4. **Lattice Points on $\ell_1$**:
   The line $\ell_1$ can be rewritten in slope-intercept form:
   \[
   y = \frac{24}{7}x - \frac{319}{7}
   \]
   Notice that $24$ and $7$ are part of the Pythagorean triple $(7, 24, 25)$. Therefore, the distance $n$ must be a multiple of $25$ for there to be lattice points on $\ell_1$.

5. **Lattice Points on $\ell_2$**:
   The line $\ell_2$ can be rewritten in slope-intercept form:
   \[
   y = \frac{12}{5}x - \frac{125}{5}
   \]
   Notice that $12$ and $5$ are part of the Pythagorean triple $(5, 12, 13)$. Therefore, the distance $n$ must be a multiple of $13$ for there to be lattice points on $\ell_2$.

6. **Common Multiples**:
   We need $n$ to be a multiple of both $25$ and $13$. The least common multiple (LCM) of $25$ and $13$ is:
   \[
   \text{LCM}(25, 13) = 325
   \]
   We need to count the multiples of $325$ that are less than $2023$:
   \[
   \left\lfloor \frac{2023}{325} \right\rfloor = 6
   \]

Conclusion:
The number of positive integer values $n$ less than $2023$ such that there exists a lattice point on both $\ell_1$ and $\ell_2$ at a distance $n$ from $(20, 23)$ is $\boxed{6}$.