A question about Integer division

Question

I'm Wondering if this equation holds for every set of positive integers $N,x_1,x_2,x_3,x_4...$

$\lfloor{\frac{N}{x_1}}\rfloor-\lfloor\frac{N}{x_1x_2}\rfloor+\lfloor\frac{N}{x_1x_2x_3}\rfloor...=\lfloor{\frac{N}{x_1}}-\frac{N}{x_1x_2}+\frac{N}{x_1x_2x_3}...\rfloor$

This conjecture has a little likelihood of validity because the difference of residues

$r_1={Nx_2x_3x_4...}\mod{x_1x_2x_3x_4...}-$

$r_2={Nx_3x_4...}\mod{x_1x_2x_3x_4...}+$

$r_3={Nx_4...}\mod{x_1x_2x_3x_4...}-$

$...$

is giving me impression to be converging to 0.

Answer 1

This does not hold for, e.g., $N=100$, $x_i=i$.

These give $$\left\lfloor{\frac{N}{x_1}}\right\rfloor-\left\lfloor\frac{N}{x_1x_2}\right\rfloor+\left\lfloor\frac{N}{x_1x_2x_3}\right\rfloor...= \left\lfloor{\frac{100}{1}}\right\rfloor - \left\lfloor{\frac{100}{2}}\right\rfloor + \left\lfloor{\frac{100}{3!}}\right\rfloor - \left\lfloor{\frac{100}{4!}}\right\rfloor = 62$$ whlle $$\left\lfloor{\frac{N}{x_1}}-\frac{N}{x_1x_2}+\frac{N}{x_1x_2x_3}...\right\rfloor= \left\lfloor{ 100\left(1-\frac{1}{e}\right)}\right\rfloor =63.$$