solve-x-x-3-3-

Question Number 182804 by mnjuly1970 last updated on 14/Dec/22

s o l v e

⌊ \underset{}{\overset{}{x}} ⌋ - ⌊ \frac{x}{3} ⌋ = 3

Answered by mahdipoor last updated on 14/Dec/22

g e t x = 3 k + j 0 ⩽ j < 3 k \in Z

⌊ x ⌋ - ⌊ x / 3 ⌋ = (3 k + ⌊ j ⌋) - (k + ⌊ j / 3 ⌋) =

2 k + ⌊ j ⌋ = 3

{\begin{cases} 0 ⩽ j < 1 \Rightarrow 2 k + 0 = 3 \Rightarrow ∄ \\ 1 ⩽ j < 2 \Rightarrow 2 k + 1 = 3 \Rightarrow k = 1 \\ 2 ⩽ j < 3 \Rightarrow 2 k + 2 = 3 \Rightarrow ∄ \end{cases}

\Rightarrow\Rightarrow 4 ⩽ x < 5

Commented by mnjuly1970 last updated on 14/Dec/22

v e r y n i c e s i r m a h d i p o o r

Answered by mr W last updated on 14/Dec/22

l e t t = \frac{x}{3}

⌊ 3 t ⌋ - ⌊ t ⌋ = 3

l e t t = n + f w i t h 0 ⩽ f < 1

3 n + ⌊ 3 f ⌋ - n = 3

⌊ 3 f ⌋ = 3 - 2 n

3 - 2 n ⩾ 0 \Rightarrow n ⩽ \frac{3}{2} \Rightarrow n ⩽ 1

3 - 2 n < 3 \Rightarrow n > 0 \Rightarrow n ⩾ 1

\Rightarrow n = 1

\Rightarrow ⌊ 3 f ⌋ = 3 - 2 = 1

\Rightarrow 1 ⩽ 3 f < 2 \Rightarrow \frac{1}{3} ⩽ f < \frac{2}{3}

\Rightarrow \frac{4}{3} ⩽ t < \frac{5}{3}

\Rightarrow \frac{4}{3} ⩽ \frac{x}{3} < \frac{5}{3}

\Rightarrow 4 ⩽ x < 5 ✓

Commented by mnjuly1970 last updated on 14/Dec/22

t h x s i r W v e r y n i c e s o l u t i o n