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.


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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

$F i r s t - d i g i t (u n i t - d i g i t) r e m o v i n g :$ $t u = 10 t + u \to \frac{(10 t + u) - u}{10} = u$ $\frac{(10 t + u) - u}{10} = \frac{10 t + u}{10 t + u} = 1$ $t = 1$ $T h e r e q u i r e d n u m b e r i s 1 u$ $11, 12, 13, . . ., 19$
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