Show that one and only one out of n,( n+2) and ( n +4) is divisble by 3, where n is any positive interger.

Show that one and only one out of n,( n+2) and ( n +4) is divisble by

1.	Show that one and only one out of n,( n+2) and ( n +4) is divisble by 3, where n is any positive interger.
Answer» When , n is dividied by 3 , let q and r be the quotient and remainder respectively . `therefore " "` n = 3q + r where , `0 le r lt 3 ` i.e., r = 0 or r =1 or r =2 (i) When , r = 0 then n = 3q , which is divisible by 3 . n + 1 = 3q +1 which leaves a remainder 1 when divided by 3. i.e., (n +1) is not divisible by 3 . n +2 = 3q +2 which leaves a remainder 2 when divided by 3 . i.e., (n +2) is not divisible by 3 . So , only n = 3q is divisible by 3 when r = 0 . `" " .... (1)` (ii) When r = 1 , then n = 3q +1 which is not divisible by 3 . `" " (because` it leaves a remainder 1) n + 1 = 3q +2 which is not divisible by 3. `" " (because` remainder= 2) n + 2 = 3q + 3 = 3(q+1) which is divisible by 3 . So , only n +2 is divisible by 3 when r= 1 . `" "....(2)` (iii) When , r=2 then n = 3q +2 which is not divisible by 3 . `" " (because ` remainder =2) `implies " " n + 1 = 3q + 3 = 3(q +1)` which is divisible by 3. n +2 = 3q + 4 = 3q + 3 + 1 = 3(q+1) + 1 which is not divisible by 3 `" " (because` remainder =1) So , only n+1 is divisible by 3 when r =2 . `" " .... (3)` Hence , from equation (1) , (2) and (3) , we can say that one and only one out of n , (n +1) and (n +2) is divisible by 3.