Question
Is 287 - 1 a Mersenne prime? If not, find one of its factors.
Answers
GPT-4o mini
To determine whether \( 287 - 1 \) is a Mersenne prime, we first calculate \( 287 - 1 \):
\[
287 - 1 = 286
\]
A Mersenne prime is defined as a prime of the form \( 2^p - 1 \) where \( p \) is a prime number. Thus, we need to express \( 286 \) as \( 2^p - 1 \) for some prime \( p \).
Rearranging gives:
\[
2^p = 287
\]
Now, let's find \( p \) by calculating the power of \( 2 \) which yields \( 287 \). Checking powers of \( 2 \):
- \( 2^8 = 256 \)
- \( 2^9 = 512 \)
Since \( 287 \) is not a power of \( 2 \), neither is it in the form \( 2^p - 1 \) for a prime \( p \), which means \( 286 \) is not a Mersenne number.
Next, we need to find the factors of \( 286 \). We can perform the factorization:
The number \( 286 \) is an even number, so we can divide by \( 2 \):
\[
286 \div 2 = 143
\]
Now we need to check if \( 143 \) is a prime number. Checking divisibility by small primes:
- It is not divisible by \( 2 \) (since \( 143 \) is odd).
- To check \( 3 \): The sum of the digits \( 1 + 4 + 3 = 8 \) is not divisible by \( 3 \).
- To check \( 5 \): The last digit is not \( 0 \) or \( 5 \).
- To check \( 7 \): \( 143 \div 7 \approx 20.4286 \) (not divisible).
- To check \( 11 \): \( 143 \div 11 = 13 \).
Since \( 143 = 11 \times 13 \), we can now write:
\[
286 = 2 \times 11 \times 13
\]
Thus, confirming that \( 286 \) is not a prime number.
In conclusion, \( 287 - 1 = 286 \) is not a Mersenne prime. One of its factors is \( 2 \).
\[
287 - 1 = 286
\]
A Mersenne prime is defined as a prime of the form \( 2^p - 1 \) where \( p \) is a prime number. Thus, we need to express \( 286 \) as \( 2^p - 1 \) for some prime \( p \).
Rearranging gives:
\[
2^p = 287
\]
Now, let's find \( p \) by calculating the power of \( 2 \) which yields \( 287 \). Checking powers of \( 2 \):
- \( 2^8 = 256 \)
- \( 2^9 = 512 \)
Since \( 287 \) is not a power of \( 2 \), neither is it in the form \( 2^p - 1 \) for a prime \( p \), which means \( 286 \) is not a Mersenne number.
Next, we need to find the factors of \( 286 \). We can perform the factorization:
The number \( 286 \) is an even number, so we can divide by \( 2 \):
\[
286 \div 2 = 143
\]
Now we need to check if \( 143 \) is a prime number. Checking divisibility by small primes:
- It is not divisible by \( 2 \) (since \( 143 \) is odd).
- To check \( 3 \): The sum of the digits \( 1 + 4 + 3 = 8 \) is not divisible by \( 3 \).
- To check \( 5 \): The last digit is not \( 0 \) or \( 5 \).
- To check \( 7 \): \( 143 \div 7 \approx 20.4286 \) (not divisible).
- To check \( 11 \): \( 143 \div 11 = 13 \).
Since \( 143 = 11 \times 13 \), we can now write:
\[
286 = 2 \times 11 \times 13
\]
Thus, confirming that \( 286 \) is not a prime number.
In conclusion, \( 287 - 1 = 286 \) is not a Mersenne prime. One of its factors is \( 2 \).