Answer to Question #89576 in Combinatorics | Number Theory for Richard landagan

Question

If n is greater than or equal to 9, and n-2 and n+2 are both primes, then 3|n.

Answer 1

There can be 3 cases : n = 3*k+1, n = 3*k+2 and n = 3*k.

Case 1: If n = 3*k+1 where k is some integer, then n+2 = 3*k+3.

n+2 > 9 so n+2 not equal 3 and n+2 divisible by 3 , then n+2 is not prime.

Case 2 : If n = 3*k+2 where k is some integer, then n-2 = 3*k.

n-2 >=7>3 and n-2 divisible by 3 , then n-2 is not prime.

The only case left

Case 3: n = 3*k , so 3|n Q.E.D.