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Step-by-Step Solution
Step 1: Identify the Triangle and Its Vertices
The triangle has vertices at
(0,0), (0,41), and (41,0). We need to count how many lattice points (points with integer coordinates) lie strictly inside this triangle (not on the boundary).
Step 2: Visualize the Lattice Points
The side connecting (0,41) and (41,0) can be represented by the line
$\, x + y = 41 \,$ (since any point on this line satisfies $\, x + y = 41 \,$).
A point $(x,y)$ lies strictly inside the triangle if:
$\, x \geq 1 \,$ and $\, y \geq 1\, $ (so that it is not on the axes $x=0$ or $y=0$),
$\, x + y < 41\, $ (so that it is below the line $\, x + y = 41\,$).
Step 3: Count the Number of Possible Integer Pairs
We proceed systematically:
When $\, x = 1,\,$ the values of $\, y \,$ can be $\, 1, 2, \ldots, 39 \,$ (since $\, 1 + y < 41 \Rightarrow y < 40\,$). That gives $\,39\,$ possible $\,y\,$ values.
When $\, x = 2,\,$ the values of $\, y \,$ can be $\, 1, 2, \ldots, 38 \,$ ($\, 2 + y < 41 \Rightarrow y < 39\,$). That gives $\,38\,$ possible $\,y\,$ values.
When $\, x = 3,\,$ the values of $\, y\,$ can be $\, 1, 2, \ldots, 37 \,$, and so on.
Finally, when $\, x = 39,\,$ the only possible $\, y\,$ is $\,1 \,$ (since $\, 39 + 1 = 40 < 41\,$), giving $\,1\,$ possible value.
Hence, we sum these counts:
$\,\displaystyle 39 + 38 + 37 + \dots + 2 + 1.\,$
Step 4: Compute the Sum
The sum of the first $\,n\,$ natural numbers is given by the formula
$\,\displaystyle \frac{n(n+1)}{2}.\,$ Here, $\,n=39,\,$ so the total number of points is
$\,\displaystyle \frac{39 \times 40}{2} = 780.\,$
Step 5: Conclude the Total Number of Interior Lattice Points
Therefore, the number of integer-coordinate points lying in the interior of the given triangle is $\,\boxed{780}.$
