show that one and only one out of p, p+2 or p+4 is divisible by 3, where p is any positive integer.

To prove one of p, p+2 or p+4 is divisible by 3.

Case 1: Let p is divisible by 3

So p = 3k for some positive integer k

Now p+2 = 3k+2 which is not divisible by 3

p+4 = 3k +3+1 =3(k+1)+1 =3m+1 not divisible by 3  

Case 2:  p+2 is divisible by 3 so

p+2 =3k

so p =3k-2  not divisible by  3

p+4 = 3k+2 = not divisible by  3

Case 3: Let p+4 is divisible by 3 so p+4 = 3k

p =3k-4=3(k-1)-1=3m-1 not divisible by 3

p+2 = 3k-2 not divisible by 3