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

Application Videos

This video explains the Method of division algorithm to find the HCF of the...

Concept Videos

This video explains the statement of Euclid's Division lemma and algorithm ...

This video explains the statement of Euclid's Division lemma and algorithm ...

This video explains the statement of Euclid's Division lemma and algorithm ...

Practice Test

Webinar

Pre Board Assessment
All Questions Ask Doubt