5 a Prove the following statements If n be an integer such than n² is even, then n² is div

12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2 = 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+ 2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.

Prove the statements: Let n be an odd positive integer. Then the sum of n consecutive integers is divisible by n.

Supposed A = {n|2 < n < 16 and n is odd} and B= {n|n is prime and 2 < n < 11}. An integer is prime if it's only divisors are 1 and the integer by itself.