关于向下取整的性质

证明：\(\lfloor{\frac{n}{xy}\rfloor}=\lfloor\frac{\lfloor{\frac{n}{x}\rfloor}}{y}\rfloor\)

设\(n=k_1x+t_1,k_1=k_2y+t_2,显然有t_1<x,t_2<y\)

\(有\lfloor\frac{\lfloor{\frac{n}{x}\rfloor}}{y}\rfloor=k_2\)

\(\lfloor{\frac{n}{xy}\rfloor}=\lfloor\frac{k_2xy+xt_2+t_1}{xy}\rfloor\)

\(=k_2+\lfloor\frac{xt_2+t_1}{xy}\rfloor\)

为了使得\(\frac{xt_2+t_1}{xy}\)的分子最大，则使\(t_1=x-1,t_2=y-1\)

\(\frac{xt_2+t_1}{xy}=\frac{xy-1}{xy}\)

证得\(\lfloor\frac{xt_2+t_1}{xy}\rfloor=0\)，故\(有\lfloor\frac{\lfloor{\frac{n}{x}\rfloor}}{y}\rfloor=k_2=\lfloor{\frac{n}{xy}\rfloor}\)

命题得证