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Floor function

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If y>0, find the range of all possible values of y such that y*floor(y) = 42.  Express your answer using interval notation.

May 12, 2022

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Suppose that $$y = r + x$$, where $$0 \leq x < 1$$ and r is an integer.

Then $$\lfloor y \rfloor = r$$.

$$r(r +x) = 42\\ x = \dfrac{42}r - r$$

Then x has to satisfy the inequality $$0 \leq x < 1$$.

We find the range of values of r such that $$0 \leq \dfrac{42}r - r < 1$$

Since y > 0, r >= 0. We multiply each term by r to get $$0 \leq 42 - r^2 < r$$.

Solving $$0 \leq 42 - r^2$$ gives $$-\sqrt{42} \leq r \leq \sqrt{42}$$.

Solving $$42 - r^2 < r$$ gives $$r > 6\text{ or }r < -7$$.

Combining the solutions of the two inequalities gives $$6 < r \leq \sqrt{42}$$. But this inequality has no integer solutions.

Therefore, there are no such value of y such that y * floor(y) = 42.

May 12, 2022