A lattice point in a plane is a point for which both coordinates are integers. If n be the number of lattice points inside the triangle whose sides are `x=0,y=0` and `9x+223y=2007` then tens place digit in n is:

Answer

To solve the problem of finding the number of lattice points inside the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( 9x + 223y = 2007 \), we can follow these steps:

### Step 1: Identify the vertices of the triangle
The triangle is formed by the following lines:
- \( x = 0 \) (the y-axis)
- \( y = 0 \) (the x-axis)
- \( 9x + 223y = 2007 \)

To find the vertices, we need to determine the points where the line intersects the axes.

1. **Intersection with the y-axis**:
Set \( x = 0 \):
\[ 9(0) + 223y = 2007 \implies y = \frac{2007}{223} = 9 \]
So, the point is \( (0, 9) \).

2. **Intersection with the x-axis**:
Set \( y = 0 \):
\[ 9x + 223(0) = 2007 \implies x = \frac{2007}{9} = 223 \]
So, the point is \( (223, 0) \).

Thus, the vertices of the triangle are \( (0, 0) \), \( (0, 9) \), and \( (223, 0) \).

### Step 2: Determine the area of the triangle
The area \( A \) of a triangle with vertices at \( (0, 0) \), \( (0, 9) \), and \( (223, 0) \) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( 223 \) (along the x-axis) and the height is \( 9 \) (along the y-axis):
\[ A = \frac{1}{2} \times 223 \times 9 = \frac{2007}{2} = 1003.5 \]

### Step 3: Count the lattice points inside the triangle
To count the number of lattice points inside the triangle, we can use Pick's Theorem, which states:
\[ I = A - \frac{B}{2} + 1 \]
where \( I \) is the number of interior lattice points, \( A \) is the area, and \( B \) is the number of boundary lattice points.

1. **Count the boundary points \( B \)**:
- On the segment from \( (0, 0) \) to \( (0, 9) \): There are \( 9 + 1 = 10 \) points.
- On the segment from \( (0, 0) \) to \( (223, 0) \): There are \( 223 + 1 = 224 \) points.
- On the segment from \( (0, 9) \) to \( (223, 0) \):
To find the number of lattice points on this line, we can use the formula for the number of lattice points on a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
\[ \text{gcd}(|x_2 - x_1|, |y_2 - y_1|) + 1 \]
Here, \( (0, 9) \) to \( (223, 0) \):
\[ \text{gcd}(223 - 0, 0 - 9) = \text{gcd}(223, 9) = 1 \implies 1 + 1 = 2 \text{ points} \]

Therefore, the total boundary points \( B \) is:
\[ B = 10 + 224 + 2 - 3 = 233 \]
(We subtract 3 because the vertices \( (0, 0) \), \( (0, 9) \), and \( (223, 0) \) were counted twice.)

2. **Calculate \( I \)**:
Now, substituting \( A = 1003.5 \) and \( B = 233 \) into Pick's theorem:
\[ I = 1003.5 - \frac{233}{2} + 1 = 1003.5 - 116.5 + 1 = 888 \]

### Step 4: Find the tens place digit of \( n \)
The number of interior lattice points \( n = 888 \). The tens place digit is \( 8 \).

### Final Answer
The tens place digit in \( n \) is \( 8 \).