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The three variables are P, V and T. n doesn't change. R is a constant.

If gases were perfectly compressible, the volume would decrease inversely proportional to the increase in pressure. There would be no reason for the temperature to increase.

But gases are not perfectly compressible. V does not decrease linearly with increases in P. In order to keep the Ideal Gas Law in balance, T has to increase.

or

Because the molecules of the gases are being mechanically compressed or being pushed together, causing them to rub tightly and trying to get them into a smaller piece of real estate than they were intended to have.

or

when you compress a gas, you have to do work on the gas to compress it...this work results in greater kinetic energy in the gas, which causes molecules to move faster, which increases the temp.

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