sir,use euclid's division lemma to show that cube of any positive integer is in the form 9M,9M+1,9M+8?

Asked by raoss1967 | 29th Nov, 2015, 12:19: PM

Expert Answer:

L e t space m space b e space a n y space p o s i t i v e space i n t e g e r space o f space t h e space f o r m space 3 q comma space 3 q plus 1 space a n d space 3 q plus 2  C a s e left parenthesis i right parenthesis r 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 comma r cubed equals 3 cubed q cubed rightwards double arrow r cubed equals 3 cubed q cubed rightwards double arrow r cubed equals 9 cross times 3 cross times q cubed rightwards double arrow r cubed equals 9 cross times 3 cross times k comma space w h e r e space k equals q cubed rightwards double arrow r cubed equals 9 m comma space w h e r e space m equals 3 k  C a s e left parenthesis i i right parenthesis r 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 comma r cubed equals open parentheses 3 q plus 1 close parentheses cubed rightwards double arrow r cubed equals open parentheses 3 cubed q cubed plus 3 cross times 3 q open parentheses 3 q plus 1 close parentheses plus 1 close parentheses rightwards double arrow r cubed equals 27 q cubed plus 9 q open parentheses 3 q plus 1 close parentheses plus 1 rightwards double arrow r cubed equals 27 q cubed plus 27 q squared plus 9 q plus 1 rightwards double arrow r cubed equals 9 open parentheses 3 q cubed plus 3 q squared plus q close parentheses plus 1 rightwards double arrow r cubed equals 9 m plus 1 space w h e r e space m equals open parentheses 3 q cubed plus q squared plus q close parentheses  C a s e left parenthesis i i i right parenthesis r 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 comma r cubed equals open parentheses 3 q plus 2 close parentheses cubed rightwards double arrow r cubed equals open parentheses 3 cubed q cubed plus 3 cross times 3 q cross times 2 cross times open parentheses 3 q plus 2 close parentheses plus 8 close parentheses rightwards double arrow r cubed equals 27 q cubed plus 54 q squared plus 36 q plus 8 rightwards double arrow r cubed equals 9 open parentheses 3 q cubed plus 6 q squared plus 4 q close parentheses plus 8 rightwards double arrow r cubed equals 9 m plus 8 space w h e r e space m equals 3 q cubed plus 6 q squared plus 4 q
S o space a n y 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 Ramamurthy | 30th Nov, 2015, 07:30: PM