To find the number of points with integral coordinates on the line segment strictly between the points A(1, 1) and B(100, 1000), we can follow these steps:
### Step 1: Determine the slope of the line
The slope \( m \) of the line passing through points A and B is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the coordinates of points A(1, 1) and B(100, 1000):
\[
m = \frac{1000 - 1}{100 - 1} = \frac{999}{99} = 11
\]
**Hint:** The slope indicates how steep the line is and helps in finding the equation of the line.
### Step 2: Write the equation of the line
Using the point-slope form of the equation of a line, we can write:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = 11 \) and point A(1, 1):
\[
y - 1 = 11(x - 1)
\]
Simplifying this, we get:
\[
y = 11x - 10
\]
**Hint:** The equation of the line helps us determine the coordinates of points on the line.
### Step 3: Identify the integral points between A and B
To find the integral points strictly between A and B, we need to find integer values of \( x \) such that \( 1 < x < 100 \). The corresponding \( y \) values will also be integers.
From the equation \( y = 11x - 10 \), we can find the integral points:
1. For \( x = 2 \), \( y = 11(2) - 10 = 12 \)
2. For \( x = 3 \), \( y = 11(3) - 10 = 23 \)
3. Continuing this way, we find the values of \( y \) for \( x = 4, 5, \ldots, 99 \).
**Hint:** The equation allows us to calculate \( y \) for each integer \( x \).
### Step 4: Count the integral points
The values of \( x \) that yield integral points must be of the form:
\[
x = 11k + 1
\]
for integers \( k \). The maximum value of \( x \) is 99, so we need to solve:
\[
1 < 11k + 1 < 100
\]
This simplifies to:
\[
0 < 11k < 99 \implies 0 < k < 9
\]
Thus, \( k \) can take values from 0 to 8, giving us 9 possible values of \( k \).
**Hint:** Counting the values of \( k \) helps us find the total number of integral points.
### Conclusion
The total number of integral points with coordinates on the line segment strictly between A(1, 1) and B(100, 1000) is:
\[
8 \text{ (from } k=1 \text{ to } k=8\text{)}
\]
Thus, the answer is **8**.
