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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

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