Show that any positive odd integer is of the form 6q+1,or 6q+3,or 6q+5,where q is some integer.

Asked by gayatrivardani2002 | 1st Jun, 2016, 05:14: PM

Expert Answer:

A c c o r d i n g space t o space E u c l i d apostrophe s space d i v i s i o n space l e m m a comma space t h e r e space e x i s t space i n t e g e r s space a comma space b comma space q space a n d space r s u c h space t h a t
a equals b q plus r comma space w h e r e space 0 less or equal than r less than b
N o w space l e t space u s space a s s u m e comma space b equals 6
T h e n space w e space h a v e comma
a equals 6 q plus r comma space w h e r e space 0 less or equal than r less than 6
T h u s comma space r space c a n space t a k e space v a l u e s space 0 comma 1 comma 2 comma 3 comma 4 comma 5

C o n s i d e r space t h e space e q u a t i o n comma space a equals 3 q plus r
C a s e 1 colon space W h e n space r equals 0
T h u s comma space a equals 6 q
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 open parentheses 3 q close parentheses
H e n c e space a space i s space a n space e v e n space n u m b e r

C a s e 2 colon space W h e n space r equals 1
T h u s comma space a equals 6 q plus 1
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 cross times 3 q plus 1
space space space equals 2 m plus 1 comma space w h e r e space m equals 3 q
H e n c e space a space i s space a n space o d d space n u m b e r

C a s e 3 colon space W h e n space r equals 2
T h u s comma space a equals 6 q plus 2
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 cross times open parentheses 3 q plus 1 close parentheses
space equals 2 m comma space w h e r e space m equals 3 q plus 1
H e n c e space a space i s space a n space e v e n space n u m b e r

C a s e 4 colon space W h e n space r equals 3
T h u s comma space a equals 6 q plus 3
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 cross times open parentheses 3 q plus 1 close parentheses plus 1
space space space equals 2 m plus 1 comma space w h e r e space m equals 3 q plus 1
H e n c e space a space i s space a n space o d d space n u m b e r

C a s e 5 colon space W h e n space r equals 4
T h u s comma space a equals 6 q plus 4
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 cross times open parentheses 3 q plus 2 close parentheses
space space space equals 2 m comma space w h e r e space m equals 3 q plus 2
H e n c e space a space i s space a n space e v e n space n u m b e r

C a s e 6 colon space W h e n space r equals 5
T h u s comma space a equals 6 q plus 5
R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
a equals 2 cross times open parentheses 3 q plus 2 close parentheses plus 1
space space space equals 2 m plus 1 comma space w h e r e space m equals 3 q plus 2
H e n c e space a space i s space a n space o d d space n u m b e r

T h e r e f o r e comma space a n y space p o s i t i v e space o d d space i n t e g e r space i s space o f space t h e space f o r m space 6 m comma space 6 m plus 1 space o r space 6 m plus 5 comma space w h e r e space m space i s space s o m e space i n t e g e r.

Answered by Vimala | 3rd Jun, 2016, 12:06: PM