Multiplying a floor function

Question

b and y are integers. Is it valid to multiply like this?

$ \left \lfloor \frac{lcm(b,y)}{b} \right \rfloor * gcd(b,y) = \left \lfloor \frac{by}{b} \right \rfloor = y$

Answer 1

Note that the least common multiple of $b$ and $y$ is a multiple of $b$, so your first fraction is an integer, and the floor function does nothing:

$$\left\lfloor\frac{\operatorname{lcm}(b,y)}b\right\rfloor=\frac{\operatorname{lcm}(b,y)}b\;.$$

Thus, you’re really just calculating

$$\frac{\operatorname{lcm}(b,y)}b\cdot\gcd(b,y)=\frac{by}b=y\;.$$