Let ( q ) be the quotient and ( r ) be the remainder when ( n ) is divided by ( 3 . ) Therefore, ( n=3 q+r, ) where ( r=0,1,2 ) ( Rightarrow quad n=3 q ) or ( n=3 q+1 ) or ( 3 q+2 ) Case(i) if ( n=3 q, ) then ( n ) is divisible by ( 3, n+2 ) and ( n+4 ) are not divisible by ( 3 . ) Case (ii) if ( n=3 q+1 ) then ( n+2=3 q+3=3(q+1) ), which is divisible by 3 and ( n+4=3 q+5, ) which is not divisible by ( 3 . ) So, only ( (n+2) ) is divisible by 3 . Case (iii) if ( n=3 q+2, ) then ( n+2=3 q+4, ) which is not divisible by 3 and ( (n+4)=3 q+6=3(q+2) ), which is divisible by 3 So, only ( (n+4) ) is divisible by ( 3 . ) Hence one and only one out of ( n,(n+2),(n+4) ) is divisible by 3
Show that one and only one out of n, n+2, n+4 is divisible by 3, where n is any positive integer.