A-number-2-2-2-find-the-number-

Question Number 10521 by ajfour last updated on 16/Feb/17

A n u m b e r (α β . . λ \dots μ 2) \times 2 = (2 α β . . λ \dots μ)

f i n d t h e n u m b e r .

Answered by mrW1 last updated on 17/Feb/17

(P a r t I)

l e t x = (α β . . λ \dots μ) = a n u m b e r w i t h n - 1 d i g i t s

l e t y = (α β . . λ \dots μ 2) = a n u m b e r w i t h n d i g i t s

l e t z = (2 α β . . λ \dots μ) = a n u m b e r w i t h n d i g i t s

y = x \times 10 + 2 = 10 x + 2

z = 2 \times 10 \dots 0 + x = 2 \times 10^{n - 1} + x

(α β . . λ \dots μ 2) \times 2 = (2 α β . . λ \dots μ)

\Leftrightarrow y \times 2 = z

(10 x + 2) \times 2 = 2 \times 10^{n - 1} + x

19 x = 2 \times 10^{n - 1} - 4 = 2 (10^{n - 1} - 2)

x = \frac{2 (10^{n - 1} - 2)}{19}

t h e f i r s t s u i t a b l e n u m b e r i s n = 18 a n d

x = \frac{2 (10^{17} - 2)}{19} = 10526315789473684

y = 105263157894736842

(105263157894736842) \times 2 = (210526315789473684)

t h e s e c o n d s u i t a b l e n u m b e r i s n = 36 a n d

x = \frac{2 (10^{35} - 2)}{19} = 10526315789473684210526315789473684

y = 105263157894736842105263157894736842

(105263157894736842105263157894736842) \times 2 = (210526315789473684210526315789473684)

t h e t h i r d s u i t a b l e n u m b e r i s n = 54 a n d

x = \frac{2 (10^{53} - 2)}{19} = 10526315789473684210526315789473684210526315789474684

y = 105263157894736842105263157894736842105263157894736842

(105263157894736842105263157894736842105263157894736842) \times 2 = (210526315789473684210526315789473684210526315789473684)

t h e g e n e r a l s o l u t i o n i s n = 18 i w i t h i = 1, 2, \dots

a n d t h e n u m b e r i s

y = 105263157894736842105263157894736842105263157894736842105263157894736842 \dots \dots

Commented by ajfour last updated on 16/Feb/17

g r e a t e x p l a n a t i o n, i c u d a r r i v e a t t h e a n s w e r b u t u r e x p l a n a t i o n i s e l a b o r a t e . b u t w a i t, i t s h o u l d e n d w i t h \dots . 736842 . t y p i n g e r r o r p r o b a b l y .

Commented by mrW1 last updated on 16/Feb/17

y o u a r e r i g h t . i h a d a t y p i n g e r r o r .

n o w {i t}^{'} s f i x e d . t h a n k y o u f o r p o i n t i n g o u t t h a t .

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