Let n be the number of integral points lying inside the parabola `y^2=8x` and circle `x^2+y^2=16`, then the sum of the digits of number n is

Answer

To solve the problem of finding the number of integral points lying inside the parabola \( y^2 = 8x \) and the circle \( x^2 + y^2 = 16 \), we will follow these steps:

### Step 1: Understand the equations
The parabola \( y^2 = 8x \) can be rewritten in the standard form as \( y^2 = 4ax \) where \( a = 2 \). This means the focus of the parabola is at the point \( (2, 0) \).

The circle \( x^2 + y^2 = 16 \) has a center at \( (0, 0) \) and a radius of \( 4 \).

### Step 2: Identify the region of interest
We need to find the integral points that lie inside both the parabola and the circle. The points must satisfy:
1. For the parabola: \( y^2 < 8x \)
2. For the circle: \( x^2 + y^2 < 16 \)

### Step 3: Analyze the symmetry
Since both the parabola and the circle are symmetric about the x-axis, we can focus on the upper half (where \( y \geq 0 \)) and then double the count of integral points found in that region.

### Step 4: Check integral points in the upper half
We will check integer values of \( x \) starting from \( x = 0 \) to \( x = 4 \) (the radius of the circle).

- **For \( x = 0 \)**:
- \( y^2 < 8(0) \) → No valid \( y \).

- **For \( x = 1 \)**:
- \( y^2 < 8(1) = 8 \) → \( y < \sqrt{8} \approx 2.83 \) → Valid \( y \): \( 0, 1, 2 \) (3 points).

- **For \( x = 2 \)**:
- \( y^2 < 8(2) = 16 \) → \( y < 4 \) → Valid \( y \): \( 0, 1, 2, 3 \) (4 points).

- **For \( x = 3 \)**:
- \( y^2 < 8(3) = 24 \) → \( y < \sqrt{24} \approx 4.89 \) → Valid \( y \): \( 0, 1, 2, 3, 4 \) (5 points).

- **For \( x = 4 \)**:
- \( y^2 < 8(4) = 32 \) → \( y < \sqrt{32} \approx 5.66 \) → Valid \( y \): \( 0, 1, 2, 3, 4, 5 \) (6 points).

### Step 5: Count the points in the upper half
Adding the valid points:
- For \( x = 1 \): 3 points
- For \( x = 2 \): 4 points
- For \( x = 3 \): 5 points
- For \( x = 4 \): 6 points

Total points in the upper half = \( 3 + 4 + 5 + 6 = 18 \).

### Step 6: Consider the lower half
Due to symmetry, the number of integral points in the lower half will also be \( 18 \).

### Step 7: Count the points on the x-axis
Now, we check the points on the x-axis:
- \( (1, 0) \), \( (2, 0) \), \( (3, 0) \) are valid points (3 points).

### Step 8: Total count of integral points
Total integral points = Points in upper half + Points in lower half + Points on x-axis
\[ n = 18 + 18 + 3 = 39 \]

### Step 9: Find the sum of the digits of \( n \)
The sum of the digits of \( 39 \) is \( 3 + 9 = 12 \).

Thus, the final answer is \( \boxed{12} \).

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