Is this identity about floor function true?

Question

When $a, b, c$ are positive integers, is this identity below is true for all $a, b, c$?

$$\left\lfloor \frac{\left\lfloor\frac ab \right\rfloor}c \right\rfloor =\left\lfloor \frac{\left\lfloor\frac ac \right\rfloor}b \right\rfloor $$

Answer 1

Using division with remainder write $a=qbc+r$ with $0\le r<bc$ and then write $r=q'b+r'$ with $0\le r'<b$, as well as $r=q''c+r''$ with $0\le r''<c$. So $a=qbc+q'b+r'=qbc+q''c+r''$. Because $0\le r<bc$ we conclude that $0\le q'<c$ and $0\le q''<b$. Then $$ \left\lfloor\frac{\left\lfloor\frac ab\right\rfloor}c\right\rfloor= \left\lfloor\frac{qc+q'}c\right\rfloor=q$$ and $$ \left\lfloor\frac{\left\lfloor\frac ac\right\rfloor}b\right\rfloor= \left\lfloor\frac{qb+q''}c\right\rfloor=q.$$ So indeed $$\left\lfloor\frac{\left\lfloor\frac ac\right\rfloor}b\right\rfloor=\left\lfloor\frac{\left\lfloor\frac ab\right\rfloor}c\right\rfloor =\left\lfloor\frac a{bc}\right\rfloor.$$