a) Always

c) When the fractional parts of \(x\) and \(y\) add up to less than one.

d) Let \(f_x\) and \(f_y\) be the fractional parts of \(x\) and \(y\) (respectively).

This will be equal to \(\lfloor x\rfloor \lfloor y\rfloor \) when \(\lfloor f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor =0\).

Extension Show that $$\lfloor x\rfloor +\left\lfloor x+\frac{1}{n}\right\rfloor +\left\lfloor x+\frac{2}{n}\right\rfloor +...+\left\lfloor x+\frac{n-1}{n}\right\rfloor =\lfloor nx\rfloor.$$ For this to be true, it is necessary (but not sufficient) that \(f_y$$\lfloor x\rfloor +\left\lfloor x+\frac{1}{n}\right\rfloor +\left\lfloor x+\frac{2}{n}\right\rfloor +...+\left\lfloor x+\frac{n-1}{n}\right\rfloor =\lfloor nx\rfloor.$$

b) When \(n$$\lfloor xy\rfloor = \lfloor (\lfloor x\rfloor +f_x)(\lfloor y\rfloor +f_x)\rfloor $$ $$=\lfloor \lfloor x\rfloor \lfloor y\rfloor +f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor $$ $$=\lfloor x\rfloor \lfloor y\rfloor +\lfloor f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor $$