CBSE Class 10 Answered

Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m ,9m+1 , or 9m+8.

Asked by gayatrivardani2002 | 01 Jun, 2016, 05:20: 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 3 T h e n space w e space h a v e comma a equals 3 q plus r comma space w h e r e space 0 less or equal than r less than 3 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 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 3 q C u b i n g space b o t h space t h e space s i d e s comma space w e space h a v e a cubed equals open parentheses 3 q close parentheses cubed space space space space equals 27 q cubed space space space space equals 9 open parentheses 3 q cubed close parentheses space space space space equals 9 m comma space w h e r e space m equals open parentheses 3 q cubed close parentheses C a s e 2 colon space W h e n space r equals 1 T h u s comma space a equals 3 q plus 1 C u b i n g space b o t h space t h e space s i d e s comma space w e space h a v e a cubed equals open parentheses 3 q plus 1 close parentheses cubed space space space space equals 27 q cubed plus 1 plus 27 q squared plus 9 q space space space space equals 9 q open parentheses 3 q squared plus 3 q plus 1 close parentheses plus 1 space space space space equals 9 m plus 1 comma space w h e r e space m equals q open parentheses 3 q squared plus 3 q plus 1 close parentheses C a s e 2 colon space W h e n space r equals 2 T h u s comma space a equals 3 q plus 2 C u b i n g space b o t h space t h e space s i d e s comma space w e space h a v e a cubed equals open parentheses 3 q plus 2 close parentheses cubed space space space space equals 27 q cubed plus 8 plus 54 q squared plus 36 q space space space space equals 9 q open parentheses 3 q squared plus 6 q plus 4 close parentheses plus 8 space space space space equals 9 m plus 8 comma space w h e r e space m equals q open parentheses 3 q squared plus 6 q plus 4 close parentheses T h e r e f o r e comma space c u b e space o f space a n y space p o s i t i v e space i n t e g e r space i s space o f space t h e space f o r m space 9 m comma space 9 m plus 1 space o r space 9 m plus 8.

Answered by Vimala | 02 Jun, 2016, 10:14: AM

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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