sir i am actually confused that how can i solve this  question the way in my textbook or in my way sir please look at this question i had also written its answer is it correct or not question isShow that a number of the form n3 - n is always divisible by 6, where n is any positive integer as follows
solution
let n be 6q .......1
 n3-n=n{n to the power 2 - 1}
now, 6q{6q to the power 2 -1} ...........using 1
6r where r is q{6q to the power 2 -1}
 thank you sir 

Asked by Monika Singh | 23rd May, 2014, 12:45: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e x p r e s s i o n space n cubed minus n : R e w r i t e space t h e space g i v e n space e x p r e s s i o n space a s : n cubed minus n equals open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses N o w space i t space i s space e n o u g h space t o space p r o v e space t h a t comma space t h e space p r o d u c t space o f space t h r e e space c o n s e c u t i v e space n u m b e r s space i s space d i v i s i b l e space b y space 6.  L e t space u s space c o n s i d e r comma space n space t o space b e space a n space o d d space n u m b e r. S o comma space n equals 2 k plus 1 comma space f o r space s o m e space i n t e g e r space k. T h u s comma space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses equals 2 k open parentheses 2 k plus 1 close parentheses open parentheses 2 k plus 2 close parentheses B y space d i v i s i o n space a l g o r i t h m space i t space i s space c l e a r space t h a t space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses space i s space a n space e v e n space n u m b e r space a n d space h e n c e space d i v i s i b l e space b y space 2.  L e t space u s space a s s u m e space t h a t space n space t o space b e space a n space e v e n space n u m b e r. S o comma space n equals 2 k comma space f o r space s o m e space i n t e g e r space k. T h u s comma space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses equals open parentheses 2 k minus 1 close parentheses 2 k open parentheses 2 k plus 1 close parentheses B y space d i v i s i o n space a l g o r i t h m space i t space i s space c l e a r space t h a t space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses space i s space a n space e v e n space n u m b e r space a n d space h e n c e space d i v i s i b l e space b y space 2.  N o w space L e t space u s space a s s u m e space t h a t space n space i s space d i v i s i b l e space b y space 3. S o comma space n equals 3 k comma space f o r space s o m e space i n t e g e r space k. T h u s comma space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses equals open parentheses 3 k minus 1 close parentheses 3 k open parentheses 3 k plus 1 close parentheses B y space d i v i s i o n space a l g o r i t h m space i t space i s space c l e a r space t h a t space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses space i s space d i v i s i b l e space b y space 3.  N o w space L e t space u s space a s s u m e space t h a t space n space i s space n o t space d i v i s i b l e space b y space 3. S o comma space n equals 3 k plus 1 comma space f o r space s o m e space i n t e g e r space k space o r space n equals 3 p plus 2 space f o r space s o m e space i n t e g e r space p T h u s comma space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses equals 3 k open parentheses 3 k plus 1 close parentheses open parentheses 3 k plus 2 close parentheses space o r o r space t h e space p r o d u c t space open parentheses n minus 1 close parentheses n open parentheses n plus 1 close parentheses equals open parentheses 3 p plus 1 close parentheses open parentheses 3 p plus 2 close parentheses open parentheses 3 p plus 3 close parentheses equals 3 open parentheses 3 p plus 1 close parentheses open parentheses 3 p plus 2 close parentheses open parentheses p plus 1 close parentheses i s space d i v i s i b l e space b y space 3 comma space b y space d i v i s i o n space a l g o r i t h m.  T h u s space t h e space p r o d u c t space i s space d i v i s i b l e space b y space 6.

Answered by Vimala Ramamurthy | 27th May, 2014, 09:28: AM