I recently took a test and if I remember the question correctly it asked what the remainder of 3^50 divided by 4 is? (it may have been 3^55)
I understand a lengthy calculation will solve the problem easily, however, what process of manipulation will get you the answer in less than a minute? I obviously don't understand large powers very well, but I know they don't expect you to actually calculate 3^50.
Such questions are very rare on the GMAT but with the current changes in the Company conducting the exam and the slight modifications in the algorithm in the CAT, they may frequent based on the test taker's level of performance.
Nevertheless; the quick fix is to observe the cyckicity of 3^n.
3^50/4 will leave a remainder of 1 while 3^55 will leave a remainder of 3. The remaiders follow a pattern if you observe and is cyclical. This will aid you solve it in less than a minute.
By the way, did this appear on a GMAT test?