Prove-that-x-2x-4x-8x-16x-32x-12345-has-no-solution-

Question Number 13389 by Tinkutara last updated on 19/May/17

Prove that

[x] + [2 x] + [4 x] + [8 x] + [16 x] + [32 x] = 12345

has no solution .

Commented by prakash jain last updated on 20/May/17

x = n + y, n \in Z^{+}, 0 ⩽ y < 1

[x] + [2 x] + [4 x] + [8 x] + [16 x] + [32 x]

= n + [y] + 2 n + [2 y] + . . + 32 n + [32 y]

= 63 n + [y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y]

= 12345

\Rightarrow n ⩽ 195

x = 195 + y

0 ⩽ y < 1

[y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y] = 60

m a x ([y]) = 0

m a x ([2 y]) = 1

m a x ([4 y]) = 3

m a x ([8 y]) = 7

m a x ([16 y) = 15

m a x ([32 y]) = 31

m a x ([y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y]) = 57

n o v a l u e o f y i s p o s s i b l e w h i c h

w i l l g i v e

[y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y] = 60

h e n c e n o s o l u t i o n

Commented by mrW1 last updated on 20/May/17

v e r y n i c e!

Commented by mrW1 last updated on 20/May/17

b u t, c a n w e f i n d a s o l u t i o n f o r

[y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y] ⩽ 57 ?

o r h o w c a n w e f i n d ?

Commented by prakash jain last updated on 20/May/17

y = . 99 or . 999

w i l l g i v e

[y] + [2 y] + [4 y] + [8 y] + [16 y] + [32 y] = 57

For other value, s a y 50

we can try various option

0 + 1 + 3 + 6 + 13 + 27 = 50

so we can pick y = . 85

[y] = 0

] 2 y] = 1

[4 y] = 3

[8 y] = 6

[16 y] = 13

[32 y] = 27

Commented by mrW1 last updated on 20/May/17

T h a n k s!

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