Question

Which number is a prime in the form  2n−1 ?(1 point)Responses 51 31 113 83

Answer 1

To determine which of the numbers \(51\), \(31\), \(113\), and \(83\) can be expressed in the form \(2n - 1\) for some integer \(n\), we rearrange this equation to find \(n\):

\[
n = \frac{x + 1}{2}
\]

where \(x\) is the number in question. For \(n\) to be an integer, \(x + 1\) must be even. This implies that \(x\) must be odd.

Now we will check each option:

1. **51**: \(51\) is odd, so we calculate \(n\):
\[
n = \frac{51 + 1}{2} = \frac{52}{2} = 26 \quad (\text{integer})
\]

2. **31**: \(31\) is odd, so we calculate \(n\):
\[
n = \frac{31 + 1}{2} = \frac{32}{2} = 16 \quad (\text{integer})
\]

3. **113**: \(113\) is odd, so we calculate \(n\):
\[
n = \frac{113 + 1}{2} = \frac{114}{2} = 57 \quad (\text{integer})
\]

4. **83**: \(83\) is odd, so we calculate \(n\):
\[
n = \frac{83 + 1}{2} = \frac{84}{2} = 42 \quad (\text{integer})
\]

All numbers \(51\), \(31\), \(113\), and \(83\) can be expressed in the form \(2n - 1\). Now we need to check if they are prime.

- **51**: Divisible by \(3\) (since \(5 + 1 = 6\)), thus not prime.
- **31**: Prime.
- **113**: Prime.
- **83**: Prime.

Thus, the numbers in the form \(2n - 1\) that are also prime are \(31\), \(113\), and \(83\). However, since we can only select one:

The correct response is **31** (the smallest valid prime in the form \(2n - 1\)).