x-2x-4x-8x-16x-32x-123456-find-all-values-of-x-for-which-this-relation-holds-

Question Number 121996 by Anuragkar last updated on 13/Nov/20

⌊ x ⌋ + ⌊ 2 x ⌋ + ⌊ 4 x ⌋ + ⌊ 8 x ⌋ + ⌊ 16 x ⌋ + ⌊ 32 x ⌋ = 123456

f i n d a l l v a l u e s o f x f o r w h i c h t h i s r e l a t i o n h o l d s ?

Answered by prakash jain last updated on 13/Nov/20

x = n + f

n (1 + 2 + 4 + \dots + 32)

⌊ 2 f ⌋ + ⌊ 4 f ⌋ + \dots + ⌋ 32 f ⌋ = 123456 - 63 n

= 63 (1959 - n) + 39

a s l h s < 63

⌊ 2 f ⌋ + ⌊ 4 f ⌋ + . . ⌊ 32 f ⌋ = 39

f ⩽ \frac{1}{2} \max lhs = 1 + 2 + 4 + 8 + 16 = 31

⌊ 2 f ⌋ = 1

⌊ 4 f ⌋ + ⌊ 8 f ⌋ + ⌊ 16 f ⌋ + ⌊ 32 f ⌋ = 38

f ⩾ \frac{3}{4} lhs > 3 + 6 + 12 + 24

\frac{1}{2} < f < \frac{3}{4}

⌊ 8 f ⌋ + ⌊ 16 f ⌋ + ⌊ 32 f ⌋ = 36

f ⩽ \frac{5}{8}

\frac{5}{8} < f < \frac{3}{4} \Rightarrow> or \frac{10}{16} < f < \frac{12}{16}

⌊ 16 f ⌋ + ⌊ 32 f ⌋ = 31

f ⩾ \frac{11}{16} \Rightarrow 11 + 22 > 31

\frac{10}{16} < f < \frac{11}{16}

[32 f ⌋ = 21

\Rightarrow f = \frac{21 + g}{32} 0 ⩽ g < 1

x = 1959 + \frac{21 + g}{32} (0 ⩽ g < 1)