Twin primes are prime numbers that are separated by 2, such as: 3 and 5, 5 and 7, 11 and 13.....etc. If you multiply any two twin primes(other that 3 and 5) and divide their product by 9, the remainder is ALWAYS 8. Give a proof that this so for all products of twin primes, with the exception above. Thank you.
The two prime numbers can be written as n+1 and n-1. Since one of the consecutive numbers n-1, n, n+1 is divisible by 3 and n-1 and n+1 are prime (so not divisible by 3 (Exception here)) so n is divisible by 3. So n*n is divisible by 9. Now: (n-1)*(n+1)=n*n-1.= -1 mod 9 = 8 mod 9.