Question

Which of the following numbers is a prime in the form 2n−1 ?(1 point)
Responses

127
127

33
33

15
15

63

Answer 1

To determine which of the given numbers is a prime in the form \(2n - 1\), we first need to check if each number can be expressed as \(2n - 1\) for some integer \(n\) and then check if that number is prime.

1. **127**:
Set \(2n - 1 = 127\).
\[
2n = 128 \implies n = 64
\]
Check if 127 is prime: The only divisors of 127 are 1 and 127, so it is prime.

2. **33**:
Set \(2n - 1 = 33\).
\[
2n = 34 \implies n = 17
\]
Check if 33 is prime: \(33 = 3 \times 11\), so it is not prime.

3. **15**:
Set \(2n - 1 = 15\).
\[
2n = 16 \implies n = 8
\]
Check if 15 is prime: \(15 = 3 \times 5\), so it is not prime.

4. **63**:
Set \(2n - 1 = 63\).
\[
2n = 64 \implies n = 32
\]
Check if 63 is prime: \(63 = 3 \times 21\), so it is not prime.

The only number among the options provided that is both in the form \(2n - 1\) and prime is **127**.