CBSE Class 10 Answered

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 | 01 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 | 03 Jun, 2016, 12:06: PM

CBSE Class 10 Answered

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CBSE Class 10 Answered

Application Videos

HCF of given two numbers

Euclid's Division Lemma

Concept Videos

Euclid's Division Lemma and Algorithm ...

Euclid's Division Lemma and Algorithm

Euclid's Division Lemma and Algorithm

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