Show that one and only one out of n,n+2 or n+4 is divisible by 3 where n is any positive integer.
To prove n , n+2 or n+4 is divisible by 3,
Case 1: Let n is divisible by 3
So n = 3k for some positive integer k
Now n+2 = 3k+2 which is not divisible by 3
n+4 = 3k +3+1 =3(k+1)+1 =3m+1 not divisible by 3
Case 2: n+2 is divisible by 3 so
n+2 =3k
so n =3k-2 not divisible by 3
n+4 = 3k+2 = not divisible by 3
Case 3: Let n+4 is divisible by 3 so n+4 = 3k
n =3k-4=3(k-1)-1=3m-1 not divisible by 3
n+2 = 3k-2 not divisible by 3