Find the number of sphere of radius `r` touching the coordinate axes.

Explanation

To find the number of spheres of radius \( r \) that touch the coordinate axes, we can analyze the situation geometrically.

### Step-by-Step Solution:

1. **Understanding the Position of the Spheres**:
Each sphere must touch all three coordinate axes: the x-axis, y-axis, and z-axis. The center of a sphere that touches these axes must be at a distance equal to its radius \( r \) from each axis.

2. **Finding the Center Coordinates**:
If a sphere of radius \( r \) is to touch the x-axis, y-axis, and z-axis, the coordinates of its center can be expressed as:
\[ (r, r, r) \]
This is because the center must be \( r \) units away from each axis.

3. **Considering Different Quadrants**:
The center of the sphere can be in different octants of the 3D coordinate system. Since the sphere can be positioned in both positive and negative directions along each axis, we can have the following combinations for the center coordinates:
- \( (r, r, r) \) (1st octant)
- \( (r, r, -r) \) (2nd octant)
- \( (r, -r, r) \) (3rd octant)
- \( (r, -r, -r) \) (4th octant)
- \( (-r, r, r) \) (5th octant)
- \( (-r, r, -r) \) (6th octant)
- \( (-r, -r, r) \) (7th octant)
- \( (-r, -r, -r) \) (8th octant)

4. **Counting the Spheres**:
Each of the above combinations represents a unique sphere that touches the coordinate axes. Since there are 8 combinations, we conclude that there are a total of 8 spheres.

### Final Answer:
Thus, the number of spheres of radius \( r \) that touch the coordinate axes is **8**.

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