To determine which numbers are prime in the form \( 2n - 1 \), we need to check if each provided number can be expressed in that form where \( n \) is an integer.
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For \( 31 \): \[ 31 = 2n - 1 \implies 32 = 2n \implies n = 16 \] Since \( 31 \) is prime.
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For \( 51 \): \[ 51 = 2n - 1 \implies 52 = 2n \implies n = 26 \] Since \( 51 \) is not prime (it equals \( 3 \times 17 \)).
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For \( 113 \): \[ 113 = 2n - 1 \implies 114 = 2n \implies n = 57 \] Since \( 113 \) is prime.
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For \( 83 \): \[ 83 = 2n - 1 \implies 84 = 2n \implies n = 42 \] Since \( 83 \) is prime.
From the evaluation:
- \( 31 \) is prime
- \( 113 \) is prime
- \( 83 \) is prime
- \( 51 \) is not prime
The numbers that are prime in the form \( 2n - 1 \) among the options are 31, 113, and 83. If asked for just one prime number, it can be any of these: 31.