Question

Suppose k is a natural number (counting number), and n=2k^2+1. Which of the following statements are true for all values of n? Choose ALL values that are true. nis odd n is even nk is divisible by 3 nis not divisible by 5

Answer 1

If k is a natural number, then n = 2k^2 + 1 is always odd and is not always divisible by 3 or not always not divisible by 5.

If we plug in a few values of k, we can see that n is always odd. For example, if k = 1, then n = 2(1)^2 + 1 = 3, which is odd. If k = 2, then n = 2(2)^2 + 1 = 9, which is also odd.

However, if we let k = 1, then n = 3, which is not divisible by 5. But if we let k = 2, then n = 9, which is also not divisible by 5. So n is not always not divisible by 5.

Similarly, if we let k = 1, then n = 3, which is not divisible by 3. But if we let k = 2, then n = 9, which is divisible by 3. So n is not always divisible by 3.