Question-let-x-lt-a-n-a-n-1-a-1-a-0-gt-N-a-0-0-amp-y-lt-a-n-a-n-1-a-1-gt-N-be-two-natural-numbers-such-that-x-y-N-find-the-number-x-

Question Number 192142 by mehdee42 last updated on 09/May/23

Q u e s t i o n

l e t x =< a_{n} a_{n - 1} \dots a_{1} a_{0} > \in N; a_{0} \neq 0 &

y =< a_{n} a_{n - 1} \dots a_{1} > \in N b e

t w o n a t u r a l n u m b e r s

s u c h t h a t \frac{x}{y} \in N

f i n d t h e n u m b e r “ x^{″} ?

Answered by AST last updated on 09/May/23

F o r a 2 - d i g i t n u m b e r, x; p o s s i b l e v a l u e s :

99, 88, 77, 66, 55, 48, 44, 39, 36, 33, 28, 26, \dots, 22, 19, \dots, 11

N o w, f o r a n n - d i g i t n u m b e r (n a t l e a s t 3)

(w h e r e a_{0} \neq 0)

\frac{x}{y} = \frac{10 x}{x - a_{0}} = \frac{10 (x - a_{0}) + 10 a_{0}}{x - a_{0}} = 10 + \frac{10 a_{0}}{x - a_{0}}

\Rightarrow x - a_{0} ∣ 10 a_{0} \Rightarrow x - a_{0} ⩽ 10 a_{0}

\Rightarrow 11 a_{0} ⩾ x

B u t t h i s i s i m p o s s i b l e s i n c e m a x {11 a_{0}} = 99

w h i c h i s n o t a t l e a s t a 3 - d i g i t n u m b e r .

\Rightarrow O n l y 2 - d i g i t s o l u t i o n s e x i s t .

Commented by mehdee42 last updated on 09/May/23

i t i s v e r y b e a u t i f u l s o l u t i o n .

i n a d d i t i o n a c c o r d i n g t o t h e c o n d i t i o n “ a_{0} \neq 0^{″}

x \neq 10, 20, \dots, 90