find a counterexample: There is no integer that has a reciprocal that is an integer

1 works: 1/1 flipped is 1/1=1

Zero.

Zero is an integer and its recriptocal is 1/0....which is 0.

Now, technically speeking, 1/0 is unknown or undefined, because you can't divide a number by 0 in math.

Read more about dividing a number by zero here: http://web2.0calc.com/questions/what-happens-when-you-divide-by-zero

Oh yeah. Anonymous's answer is probably more accurate. 

In fact, 1 and -1 are the only two things that do work....!!!!