Question-191775

Question Number 191775 by Mingma last updated on 30/Apr/23

Answered by AST last updated on 30/Apr/23

\overset{____________}{A B C D E F} = 10000 \overset{_______}{A B C D} + \overset{___}{E F}

17 [10000 \overset{___}{A B} + \overset{_______}{C D E F}] = 12 [100 \overset{_______}{C D E F} + \overset{___}{A B}]

170000 \overset{___}{A B} - 12 \overset{___}{A B} = 1200 \overset{_______}{C D E F} - 17 \overset{________}{C D E F}

\Rightarrow \frac{\overset{___}{A B}}{\overset{________}{C D E F}} = \frac{1183}{169988} = \frac{13}{1868}

N o t e t h a t \overset{___}{A B} a n d \overset{_______}{C D E F} c a n s t i l l b e i n c r e a s e d

b y a f a c t o r o f 5 w h i l e \overset{___________}{A B C D E F} r e m a i n s a

6 - d i g i t n u m b e r

\Rightarrow T h e l a r g e s t 6 - d i g i t n u m b e r = 659340

T h e s m a l l e s t 6 - d i g i t n u m b e r = 131868

Commented by Mingma last updated on 30/Apr/23

Excellent!

Answered by Frix last updated on 30/Apr/23

This leads to

\frac{A B}{13} = \frac{C D E F}{1868}

With all digits different we have only 2

possibilities :

\frac{39}{13} = \frac{5604}{1868} = 3

\frac{65}{13} = \frac{9340}{1868} = 5

\Rightarrow

Answer is 659340

Commented by Mingma last updated on 30/Apr/23

Excellent!