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[Original answer]

Probably it would travel a bit faster than on Earth, since the atmosphere of Mars is thinner than Earth's.

[Revised Answer]

The chemical energy in the charge of explosive in a bullet is fixed; the mass of the lead in the actual projectile is fixed.

All things being equal:

A fixed amount of energy will impart a specific velocity to a fixed mass. The mass will absorb/utilize the energy in the same way and the speed should remain the same. The only variant is the density of the atmospheric gas ahead of the bullet in the barrel of the gun offering "resistance" to the passage of the bullet. What happens after it leaves the barrel is irrelevant since other forces begin to work upon it and change the speed. Specifically, it will slow down more quickly in Earth's atmosphere, but the initial speed upon leaving the barrel will be virtually the same. If the question said will it be faster or slower 100 metres after it leaves the gun barrel then the answer is clear (faster on Mars; slower on Earth)

Mars' average atmospheric pressure is 0.6 pounds per square inch; Earth's is 15 pounds per square inch. Earth's atmospheric density is about 25 times greater. You might think this would make a substantial difference, but no, it is all but immeasurable... The pressure within the firing chamber that transfers the energy from the chemical explosion to the motion of the lead projectile reaches an astonishing 6,000 psi and in some special cases can exceed 30,000 psi in a small gun. The difference between 15psi and 0.6psi is dwarfed into insignificance compared to 6000 or more.

So the answer is that is would travel a tiny, tiny, tiny bit faster but it would be virtually undetectable.

On the other hand, it would travel much farther if fired in a way not to encounter another object. Fired straight up on Earth it would need to overcome our gravity with an acceleration in the opposite direction of ~9.8 metres per second; on Mars this would be only ~3.7m/s, or about 1/3, so it would decelerate much more slowly, and consequently travel ~3 times the distance.

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