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 | 1st 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 | 2nd Jun, 2016, 10:14: AM