Question

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

Answer 1

To determine which numbers in the form \(2n - 1\) are prime, we first recognize that \(2n - 1\) produces odd numbers for integer values of \(n\).

Next, let's check each of the provided numbers to see which can be expressed as \(2n - 1\):

1. **31**:
\[
31 = 2n - 1 \implies 2n = 32 \implies n = 16
\]
31 is prime.

2. **113**:
\[
113 = 2n - 1 \implies 2n = 114 \implies n = 57
\]
113 is prime.

3. **51**:
\[
51 = 2n - 1 \implies 2n = 52 \implies n = 26
\]
51 is not prime (divisible by 3 and 17).

4. **83**:
\[
83 = 2n - 1 \implies 2n = 84 \implies n = 42
\]
83 is prime.

The prime numbers in the form \(2n - 1\) from the given options are 31, 113, and 83.

Thus, the answer is that all **31, 113, and 83** are prime numbers in the form \(2n - 1\).