Which property is constant for all parts of this spectrum? Why

The electromagnetic spectrum includes radio waves, infrared waves, visible light, ultraviolet waves, and X-rays.

A. Frequency
B. Period
C. Speed
D. Wavelength

C. They all travel at the speed of light in a vacuum.

These various forms of radiation differ only in their frequency, but they all travel at the same speed -- "c", the velocity of light in vacuum. Since period is 1/frequency, and since wavelength is c/frequency, both of those properties also vary with frequency. The only constant property they all share is their speed. So... (C) is the answer, and "c" is the speed.

The frequencies, wavelengths and periods of the electromagnetic spectrum will vary from one end to the other.
The electromagnetic spectrum has constant speed provided that the medium of travel is constant. The speed of light will vary for different translucent media. This is demonstrated by the properties of lenses where the light path is bent and the velocity has a different direction as a consequence.