Given:
- We have a rectangular parallelepiped.
- The volume of the parallelepiped constructed on the diagonals of the faces of the given parallelepiped is m times the volume of the given parallelepiped.

To find:
The value of m.

Solution:
Let's consider the given rectangular parallelepiped with dimensions a, b, and c.

Step 1: Finding the Diagonals of the Faces
The diagonals of the faces of the given parallelepiped can be found using the Pythagorean theorem.

- Diagonal of the first face = √(a² + b²)
- Diagonal of the second face = √(b² + c²)
- Diagonal of the third face = √(a² + c²)

Step 2: Constructing the Parallelepiped on the Diagonals
To construct the parallelepiped on the diagonals, we need to find the lengths of the three edges sharing a common vertex.

- Edge 1 = √(a² + b²) (Diagonal of the first face)
- Edge 2 = √(b² + c²) (Diagonal of the second face)
- Edge 3 = √(a² + c²) (Diagonal of the third face)

Step 3: Finding the Volume of the Parallelepiped
The volume of a parallelepiped can be found by taking the dot product of any two edges and their cross product magnitude.

- Volume of the given parallelepiped = |a · b x c|

Step 4: Finding the Volume of the Parallelepiped Constructed on the Diagonals
The volume of the parallelepiped constructed on the diagonals can be found using the same method as above.

- Volume of the parallelepiped on the diagonals = |(√(a² + b²)) · (√(b² + c²)) x (√(a² + c²))|

Step 5: Evaluating m
We are given that the volume of the parallelepiped constructed on the diagonals is m times the volume of the given parallelepiped.

Therefore, m = (Volume of the parallelepiped on the diagonals) / (Volume of the given parallelepiped)

m = |(√(a² + b²)) · (√(b² + c²)) x (√(a² + c²))| / |a · b x c|

Simplifying the expression, we get:

m = (√(a² + b²)) · (√(b² + c²)) · (√(a² + c²)) / |a · b x c|

Since the volume of a parallelepiped is always positive, we can remove the absolute value signs.

m = (√(a² + b²)) · (√(b² + c²)) · (√(a² + c²)) / (a · b x c)

Step 6: Evaluating m for a Rectangular Parallelepiped
For a rectangular parallelepiped, the dot product of the edges is equal to the product of their magnitudes.

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