((⌊(x/3) ⌋)/(⌊ (x/4) ⌋)) = ((21)/(16)) ; x=?


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Question Number 216076 by efronzo1 last updated on 27/Jan/25

$\frac{⌊ \frac{x}{3} ⌋}{⌊ \frac{x}{4} ⌋} = \frac{21}{16}; x = ?$
	Answered by mr W last updated on 28/Jan/25

	$⌊ \frac{x}{3} ⌋ = 21 k$ $⌊ \frac{x}{4} ⌋ = 16 k \neq 0 \Rightarrow k \neq 0$ $\frac{x}{3} = 21 k + a w i t h 0 ⩽ a < 1$ $\Rightarrow x = 63 k + 3 a$ $\frac{x}{4} = 16 k + b w i t h 0 ⩽ b < 1$ $\Rightarrow x = 64 k + 4 b$ $63 k + 3 a = 64 k + 4 b$ $- 4 < k = 3 a - 4 b < 3$ $\Rightarrow - 3 ⩽ k ⩽ 2 \land k \neq 0$ $21 k ⩽ \frac{x}{3} < 21 k + 1$ $3 \times 21 k ⩽ x < 3 \times 21 k + 3$ $16 k ⩽ \frac{x}{4} < 16 k + 1$ $4 \times 16 k ⩽ x < 4 \times 16 k + 4$ $k = - 3 :$ $x \in [- 189, - 186)$ $x \in [- 192, - 188)$ $\Rightarrow - 189 ⩽ x < - 188 ✓$ $k = - 2 :$ $x \in [- 126, - 123)$ $x \in [- 128, - 124)$ $\Rightarrow - 126 ⩽ x < - 124 ✓$ $k = - 1 :$ $x \in [- 63, - 60)$ $x \in [- 64, - 60)$ $\Rightarrow - 63 ⩽ x < - 60 ✓$ $k = 1 :$ $x \in [63, 66)$ $x \in [64, 68)$ $\Rightarrow 64 ⩽ x < 66 ✓$ $k = 2 :$ $x \in [126, 129)$ $x \in [128, 132)$ $\Rightarrow 128 ⩽ x < 129 ✓$
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