solve-1-x-3-x-4-

Question Number 155951 by mnjuly1970 last updated on 06/Oct/21

s o l v e :

⌊ \frac{1}{x} ⌋ + ⌊ \frac{3}{x} ⌋ = 4

Answered by mr W last updated on 06/Oct/21

l e t t = \frac{1}{x}

⌊ t ⌋ + ⌊ 3 t ⌋ = 4

l e t t = n + f w i t h 0 ⩽ f < 1

n + 3 n + ⌊ 3 f ⌋ = 4

4 n + ⌊ 3 f ⌋ = 4

c a s e 0 ⩽ f < \frac{1}{3} :

4 n + 0 = 4 \Rightarrow n = 1 \Rightarrow 1 ⩽ t < \frac{4}{3}

c a s e \frac{1}{3} ⩽ f < \frac{2}{3} :

4 n + 1 = 4 \Rightarrow n o s o l u t i o n!

c a s e \frac{2}{3} ⩽ f < 1 :

4 n + 2 = 4 \Rightarrow n o s o l u t i o n!

o n l y s o l u t i o n i s 1 ⩽ t < \frac{4}{3}

i . e . 1 ⩽ \frac{1}{x} < \frac{4}{3}

\Rightarrow \frac{3}{4} < x ⩽ 1

Commented by mnjuly1970 last updated on 06/Oct/21

g r a t e f u l s i r W