find-all-such-numbers-if-we-make-its-last-digit-say-k-as-its-first-digit-the-number-becomes-k-times-large-as-before-k-k-k-k-

Question Number 103888 by mr W last updated on 18/Jul/20

f i n d a l l s u c h n u m b e r s :

i f w e m a k e i t s l a s t d i g i t, s a y k, a s i t s

f i r s t d i g i t, t h e n u m b e r b e c o m e s k

t i m e s l a r g e a s b e f o r e .

(\dots k) \to (k \dots) = k \times (\dots k)

Answered by MAB last updated on 18/Jul/20

x = \underset{i = 0}{\sum^{n}} 10^{i} a_{i} w h e r e a_{i} \in {0, 1 \dots 9} a n d a_{0} = k

x^{'} = 10^{n} k + \underset{i = 0}{\sum^{n - 1}} 10^{i} a_{i + 1} = k x

h e n c e

10^{n} k + \underset{i = 0}{\sum^{n - 1}} 10^{i} a_{i + 1} = k^{2} + \underset{i = 0}{\sum^{n - 1}} 10^{i + 1} a_{i + 1}

(10^{n} - k) k = 9 \underset{i = 0}{\sum^{n - 1}} 10^{i} a_{i + 1}

(10^{n} - k) k = 9 k x

10^{n} - k = 9 x

k = 1 a n d x = 111 \dots 1

Commented by mr W last updated on 18/Jul/20

t h a n k s s i r!

Answered by mr W last updated on 18/Jul/20

l e t x = \underset{↽ n d i g i t s ⇁}{\dots} ⩽ 10^{n} - 1

N = \underset{↽ n d i g i t s ⇁}{\dots} k = 10 x + k

k \underset{↽ n d i g i t s ⇁}{\dots} = 10^{n} k + x

10^{n} k + x = k (10 x + k)

\Rightarrow x = \frac{k (10^{n} - k)}{10 k - 1} ⩽ 10^{n} - 1

c a s e k = 1 :

x = \frac{10^{n} - 1}{9} = \underset{↽ n d i g i t s ⇁}{111 . . 111}

x = \frac{10^{n} - 1}{9} = \underset{↽ n d i g i t s ⇁}{111 . . 111}

\Rightarrow N = \underset{↽ n d i g i t s ⇁}{111 . . 1111} w i t h n \in N

c a s e k = 2 :

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

w e c a n f i n d

n = 18 m - 1 w i t h m \in N

e x a m p l e n = 17 :

x = 10526315789473784

\Rightarrow N = 105263157894737842

c h e c k :

2 \times 105263157894737842

= 210526315789473784

e x a m p l e n = 35 :

x = 10526315789473784210526315789473784

\Rightarrow N = 105263157894737842105263157894737842

c h e c k :

2 \times 105263157894737842105263157894737842

= 210526315789473784210526315789473784

(t o b e c o n t i n u e d)

Commented by mr W last updated on 18/Jul/20

c a s e k = 3 :

h e r e {w e}^{'} l l t r y a n o t h e r w a y t o f i n d

t h e n u m b e r x .

n o t e : i n f o l l o w i n g t h e d i g i t s o f t h e

n u m b e r a r e d i s p l a y e d i n v e r s e l y, i . e .

t h e l a s t d i g i t i s t h e l e f t m o s t .

39731427155698602685728443013

\times 3 97314271556986026857284430139

∣\leftarrow l a s t d i g i t f i r s t d i g i t \to∣

\Rightarrow x = 103448275862068965517241379

\Rightarrow N = 1034482758620689655172413793

c h e c k :

3 \times 1034482758620689655172413793

= 3103448275862068965517241379

w e s e e t h e d i g i t s o f n u m b e r N m a y b e

r e p e a t e d a n y t i m e s . s o w e h a v e

i n f i n i t e n u m b e r s w h i c h s a t i s f y t h e

r e q u i r e m e n t .

Commented by mr W last updated on 18/Jul/20

c a s e k = 4 :

u s i n g t h e s a m e m e t h o d a s b e f o r e,

4652014

\times 4 6520146

\Rightarrow x = 10256

\Rightarrow N = 102564

c h e c k :

4 \times 102564

= 410256

s i m i l a r l y t h e d i g i t s o f N m a y b e

r e p e a t e d a n y t i m e s . f o r e x a m p l e

N = 102564102564102564

Commented by mr W last updated on 18/Jul/20

c a s e k = 5 :

x = \frac{k (10^{n} - k)}{10 k - 1} = \frac{5 (10^{n} - 5)}{7 \times 7}

i t s e e m s t o h a v e n o s o l u t i o n .

Commented by mr W last updated on 19/Jul/20

c a s e k = 6 :

x = \frac{k (10^{n} - k)}{10 k - 1} = \frac{6 (10^{n} - 6)}{59}

i t s e e m s t o h a v e n o s o l u t i o n .

Facebook Tweet Pin