Must a continuous function -LSB- math -RSB- f : \ mathbb -LCB- R -RCB- ^ n \ to \ mathbb -LCB- R -RCB- -LRB- n > 1 -RRB- -LSB- / math -RSB- , such that its integral along any straight line is -LSB- math -RSB- 0 -LSB- / math -RSB- , be -LSB- math -RSB- 0 -LSB- / math -RSB- everywhere ?
How do I prove that when -LSB- math -RSB- f -LRB- x -RRB- -LSB- / math -RSB- is divided by -LSB- math -RSB- -LCB- x -RCB- ^ -LCB- 3 -RCB- - x -LSB- / math -RSB- , the remainder is of the form -LSB- math -RSB- 3r -LRB- x -RRB- -LSB- / math -RSB- , where -LSB- math -RSB- r -LRB- x -RRB- -LSB- / math -RSB- is a polynomial with integer coefficients and it is given that -LSB- math -RSB- f -LRB- x -RRB- -LSB- / math -RSB- is a polynomial with integer coefficients and -LSB- math -RSB- f -LRB- n -RRB- -LSB- / math -RSB- is divisible by -LSB- math -RSB- 3 -LSB- / math -RSB- for all integers -LSB- math -RSB- n -LSB- / math -RSB- ?