Question

Show that if n is an integer and n³ + 5 is odd, then n is even using

Answer 1

To show that if n is an integer and n³ + 5 is odd, then n is even, we can use a proof by contradiction.

Step-by-step explanation:

To show that if n is an integer and n³ + 5 is odd, then n is even, we can use a proof by contradiction.

1. Assume that n is odd. Then we can write n as 2k+1, where k is an integer.
2. Substituting this value of n into the expression n³ + 5, we get (2k+1)³ + 5.
3. Expanding this expression, we have (8k³ + 12k² + 6k + 1) + 5 = 8k³ + 12k² + 6k + 6.
4. As we can see, the expression 8k³ + 12k² + 6k + 6 is even, therefore n³ + 5 cannot be odd if n is odd.
5. This contradicts our initial assumption, so n cannot be odd. Therefore, n must be even.