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# Floor function

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How many values of r are there such that $\lfloor r/4 \rfloor + r = 15.5?$

May 10, 2022

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Moving r to the other side gives $$\left\lfloor\dfrac r4\right\rfloor = 15.5 -r$$.

Then 15.5 - r is an integer. Then r = k + 0.5 for some integer k.

Then $$\left\lfloor\dfrac {k + \frac12}4\right\rfloor = 15 - k$$.

That means $$15-k\leq \dfrac{k + \frac12}4 < 16 - k$$ by definition of floor function. Now it becomes a compound inequality. To proceed, solve the compound inequality and list its integer solutions.

May 10, 2022