Answer all questions please. I will give a good review

Question 1 Which of the following is the correct definition of an even integer? An integer p is even if and only if it can be written in the form p = 2p An integer p is even if and only if it can be written in the form p = 2k + 1 for some integer k An integer p is even if and only if it can be written in the form p = 2k for some integer k An integer p is even if and only if it can be written in the form p = 4k for some integer k Question 3 Let a be a non-zero integer and suppose that a divides b (this is written as alb), then which of the following statements is TRUE? a= kb for some integer k Ek ez, b= ka O a is a divisor or factor of b a is a multiple of b b= ka for some integer k. For example, 4 divides 24 since 24 = k. 4 (in this case k = 6, which is an integer). Question 6 Consider the statement If two integers have different parities, then their sum is odd. Which of the following provides a correct, complete and logical proof of the above statement? O Proof IV Let a=5 and 58. a+b=5+8 13 which is odd. Let a= 2 and b=3. a + b = 2 + 3 = 5 which is odd. Therefore the sun is always odd. o Proof III Suppose that a is even and b is odd. 3p, qe Z such that a=2p and b=2411. The sum a+b = 2p+2q+1 = 2(+1. ( Z. . Therefore the sum is odd. Proof I Suppose the sum of two integers a and b is odd. . The sum of two even numbers is even; the sum of two odd numbers is even the sum of an even and an odd number is odd. Therefore one of a or b (but not both) has to be odd, and the other even. Hence a and i have different parities. Proof 11 Suppose a and b are two integers with different parities. Without loss of generality, assume that a is even and b is odd. 3pq e Z such that a = 2p and b = 2q+1. The sum a+b = 2p+24+1 = 21+1,1 Z. Therefore a + b is odd