show that

suppose pis divisible by 3

 then p=3q (say) where q is some non negative integer.

the next multiple of 3 will be only p+3,p+6 etc( multiples of 3 will occur only after a gap pf 3)

So, if p is a multiple of 3 then p+2 and p+4 can't be multiples of 3

Next if p+2 is a multiple of 3 then p+2= 3r(say)

So, the multiple of 3 before p+2 must be (p+2)-3 and the next mutiple after p+2 will be (p+2)+3

i.e. p-1 and p+5 respectively. So if p+2 is a multiple of 3 then por p+4 can't be multiples of 3.

 By a similar logice, we can show  that  if p+4 is a multiple of 3 then p or p+2 can't be multiples of 3.

 This explains the answer.