Select three distinct digits. Form a three digit number. Form the other number by reversing the digits. Find the difference of these two numbers. What is the GCD of all such differences? Please give step-by-step detailed answer.

Asked by Divyaa | 12th Dec, 2015, 09:31: PM

Expert Answer:

L e t space x y z space b e space t h e space r e q u i r e d space n u m b e r x y z equals 100 x plus 10 y plus z... left parenthesis 1 right parenthesis R e v e r s n g space t h e space d i g i t s comma space w e space h a v e comma z y x equals 100 z plus 10 y plus x... left parenthesis 2 right parenthesis A s s u m i n g space x greater than z comma space s u b t r a c t i n g space e q u a t i o n space left parenthesis 2 right parenthesis space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e comma D i f f e r e n c e space o f space t w o space n u m b e r s equals 100 open parentheses x minus 1 minus z close parentheses plus 90 plus open parentheses 10 plus z minus x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals T h u s space t h e space d i g i t space i n space t h e space h u n d r e d s space p l a c e space i s space open parentheses x minus 1 minus z close parentheses space a n d space t h e space d i g i t space i n space t h e space u n i t s space p l a c e space i s space open parentheses 10 plus z minus x close parentheses A d d space t h e space d i g i t space i n space t h e space h u n d r e d s space p l a c e space t o space t h e space d i g i t space i n space t h e space u n i t s space p l a c e. R e s u l t equals x minus 1 minus z plus 10 plus z minus x equals 9 D i g i t space i n space t h e space t e n s space p l a c e space o f space t h e space d i f f e r e n c e space o f space t w o space n u m b e r s space i s space a l w a y s space 9. N o w space s u b t r a c t space t h e space d i g i t space i n t h e space 10 s space p l a c e comma space t h a t space i s space 9 space f r o m space t h e space r e s u l t space o f space a d d i t i o n space o f space h u n d r e d apostrophe s space d i g i t space a n d space t h e space u n i t s space d i g i t. T h a t space i s space 9 minus 9 equals 0 rightwards double arrow D i f f e r e n c e space o f space t w o space n u m b e r s space i s space d i v i s i b l e space b y space 11  A l s o space t h e space s u m space o f space a l l space t h e space d i g i t s space o f space t h e space d i f f e r e n c e space o f space t w o space n u m b e r s space i s space 9. H e n c e space t h e d i f f e r e n c e space i s space d i v i s i b l e space b y space 9 S o space t h e space d i f f e r e n c e space i s space d i v i s i b l e space b y space b o t h space 9 space a n d space 11. T h a t space i s space t h e space d i f f e r e n c e space o f space t w o space n u m b e r s space i s space d i v i s i b l e space b y space 99.  T h u s comma space t h e space G C D space o f space t h e space d i f f e r e n c e space o f space t w o space n u m b e r s space i s space 99.

Answered by Vimala Ramamurthy | 14th Dec, 2015, 10:00: AM