Find-a-number-that-becomes-N-times-smaller-if-the-first-digit-is-removed-in-the-following-cases-1-N-17-2-N-27-3-N-37-4-N-47-

Question Number 99058 by Learner-123 last updated on 18/Jun/20

F i n d a n u m b e r t h a t b e c o m e s N t i m e s

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

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

1) N = 17

2) N = 27

3) N = 37

4) N = 47 .

Commented by Rasheed.Sindhi last updated on 18/Jun/20

1) N = 17

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

i {d o n}^{'} t u n d e r s t a n d w h a t i s m e a n t .

s a y t h e n u m b e r i s X Y Y Y Y Y .

i t s h o u l d b e

Y Y Y Y Y = \frac{X Y Y Y Y Y}{N} w i t h N = 17 ?

Commented by Rasheed.Sindhi last updated on 18/Jun/20

S i r I u n d e r s t a n d b y f i r s t d i g i t

^{'} u n i t - {d i g i t}^{'} . S o i f r e m o v e 7 f r o m

17 i t b e c o m e s 1 w h i c h i s 17 t i m e s

s m a l l e r t h a n 17 .

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

t h a n k s s i r!

(X y) = 10 X + y = 17 X

\Rightarrow y = 7 X ⩽ 9

\Rightarrow X ⩽ \frac{9}{7} \Rightarrow X = 1 \Rightarrow y = 7

\Rightarrow t h e n u m b e r i s 17 .

(X y) = 10 X + y = 27 X

\Rightarrow y = 17 X ⩽ 9 \Rightarrow X ⩽ \frac{9}{17} \Rightarrow n o s o l u t i o n

Commented by Learner-123 last updated on 18/Jun/20

t h a n k s b o t h s i r s!

Commented by Rasheed.Sindhi last updated on 18/Jun/20

F i r s t - d i g i t (u n i t - d i g i t) r e m o v i n g :

t u = 10 t + u \to \frac{(10 t + u) - u}{10} = u

\frac{(10 t + u) - u}{10} = \frac{10 t + u}{10 t + u} = 1

t = 1

T h e r e q u i r e d n u m b e r i s 1 u

11, 12, 13, \dots, 19