Question

Put the following statements into order to prove that if 3n+4 is even then n is even. Put N next to the 3 statements that should not be used.1. Then, 3n+4=3(2k+1)+4=6k+7=2(3k+3)+1.2. By definition of odd, there exists integer k such that n=2k+1.3. Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even, .4. Therefore, by definition of odd, 3n+4 is odd.5. Since k is an integer, t =3k+3 is also an integer.6. Thus, there exists an integer t such that 3n+4=2t+1.7. Suppose n is odd.8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even, .9. Suppose 3n+4 is even.10. By definition of odd, n=2k+1.

Answer 1

Answer: 7, 2, 1, 5, 6, 4, 8

Explanation:

1. Then, 3n+4=3(2k+1)+4=6k+7=2(3k+3)+1.

2. By definition of odd, there exists integer k such that n=2k+1.

N 3. Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even, .

4. Therefore, by definition of odd, 3n+4 is odd.

5. Since k is an integer, t =3k+3 is also an integer.

6. Thus, there exists an integer t such that 3n+4=2t+1.

7. Suppose n is odd.

8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even, .

N 9. Suppose 3n+4 is even.

N 10. By definition of odd, n=2k+1.

Right order:

7. Suppose n is odd.

2. By definition of odd, there exists integer k such that n = 2k+1.

1. Then, 3n+4 = 3(2k+1)+4 = 6k+7 = 2(3k+3)+1.

5. Since k is an integer, t = 3k+3 is also an integer.

6. Thus, there exists an integer t such that 3n+4 = 2t+1.

4. Therefore, by definition of odd, 3n+4 is odd.

8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even.