x-3-x-4-21-16-x-

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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