Define an integer k to be odd if k 1 is even. Write up a formal proof of the following, using an indirect proof: Claim: For any natural numbers m and n, if m is odd and n is odd, then m n is even. (Notice--this is not hard to prove, but make sure that you are using a proof by contradiction. And use the official definitions of even and odd.)

"for all natural numbers m, n, p, we have that if p | (m * n) then either p | m or p | n" Try different concrete values for m, n, and p. Is this a true statement or a false statement? If false, give a counterexample. If true, find a proof.

- A Pythagorean prime number is a prime number of the form 4n + 1, where n ≥ 1. A Mersenne prime number is a prime number of the form 2" – 1. A perfect number is a number that is equal to the sum of all of its divisors, including 1 but excluding the number itself. Consider the following statements: (1) Let m is the smallest Pythagorean prime number. Then, 2¹ – 1 is a Mersenne prime number. (2) If m is a perfect number, then 2 – 1 is a Mersenne prime number. Select one of the following choices: (a) (1) is True and (2) is False. (c) (1) and (2) are True. (b) (1) is False and (2) is True. (d) (1) and (2) are False.

Give a carefully worded proof for the following statements using direct proof, contrapositive proof or proof by contradiction. No matter which proof method you choose to use, make sure you follow the outline properly. (If-and-Only-If Proof) Given an integer a and b, then b2(a + 3) is even if and only if a is odd or b is even. (Use the basic definitions of even and odd in your proof.)

In each of the following statements, n represents a positive integer. One of the statements is true and the other two are false.  If n is even, then n is not prime  if n is a square number, then n is not prime  if n is a prime number and n ≥ 10, then n+2 is prime or n+4 is prime.  A. What two statements are false and in each case give a counter example to show that it is false  B. Prove the true statement

Use the PMI to prove that 3 - 1 is even for all n E N. Proof. Base case: Since 3¹ - 1 = 2, which is even. Thus the statement is true for n = [Select] Inductive step: Assume that there is a natural number n such that 3 - 1 is even. Then 3-1 = [Select] for some integer k. Then 3" [Select] ✓. On both sides, first multiply by 3, then subtract 1, and simplify to get 3+1 -1 = 2( [Select] which is an [Select] integer. Hence, by the PMI, 3 - 1 is even for all n E N. )

Give a direct proof: Let a, b, and c be integers. If a | b and a c, then a | (b. c). Remember that you must use the definition of | in your proof. Suppose a, b, and care integers and a | b and a | c. By the definition of 1, b = ak₁ and c = for some integers k₁ and k₂. Therefore, b . c = a, and is an integer, so a | b. c.

Prove the following statements: (a) If n is a positive integer then n 2 + 1 is an odd positive integer. (b) If n 2 + 1 is an even positive integer, and n is a positive integer then n is an odd positive integer.