x-2x-3x-6-x-

Question Number 166257 by mnjuly1970 last updated on 16/Feb/22

⌊ x ⌋ ⌊ 2 x ⌋ ⌊ 3 x ⌋ = 6

x = \overset{}{?}

Commented by MJS_new last updated on 16/Feb/22

1 ⩽ x < \frac{4}{3}

Commented by mnjuly1970 last updated on 17/Feb/22

✓ ✓ t h a n k y o u

Answered by mr W last updated on 17/Feb/22

s a y x = n + f w i t h 0 ⩽ f < 1

6 = ⌊ x ⌋ ⌊ 2 x ⌋ ⌊ 3 x ⌋ ⩾ n \times 2 n \times 3 n = 6 n^{3}

\Rightarrow n^{3} ⩽ 1 \Rightarrow n ⩽ 1

6 = ⌊ x ⌋ ⌊ 2 x ⌋ ⌊ 3 x ⌋ < n \times (2 n + 2) \times (3 n + 3) = 6 n^{3} + . . + 6

\Rightarrow n > 0

\Rightarrow n = 1

1 \times (2 + ⌊ 2 f ⌋) \times (3 + ⌊ 3 f ⌋) = 6

6 + 3 ⌊ 2 f ⌋ + 2 ⌊ 3 f ⌋ + ⌊ 2 f ⌋ ⌊ 3 f ⌋ = 6

3 ⌊ 2 f ⌋ + 2 ⌊ 3 f ⌋ + ⌊ 2 f ⌋ ⌊ 3 f ⌋ = 0

\Rightarrow ⌊ 2 f ⌋ = 0, ⌊ 3 f ⌋ = 0

\Rightarrow 0 ⩽ 3 f < 1 \Rightarrow 0 ⩽ f < \frac{1}{3}

\Rightarrow s o l u t i o n i s 1 + 0 ⩽ x < 1 + \frac{1}{3}, i . e .

1 ⩽ x < \frac{4}{3}

Commented by mnjuly1970 last updated on 17/Feb/22

t h a n k s a l o t s i r W