Question
Determine whether 255−1
is a Mersenne prime. If not, find a factor of 255−1
.(1 point)
Responses
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 222−1
.
No, 2 superscript 55 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 55 baseline minus 1 is 2 superscript 22 baseline minus 1 .
Yes, 255−1
is a Mersenne prime.
Yes, 2 superscript 55 baseline minus 1 is a Mersenne prime.
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 210+29+28+27+26+25+24+23+22+2+1=2,047
.
No, 2 superscript 55 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 55 baseline minus 1 is 2 superscript 10 baseline plus 2 superscript 9 baseline plus 2 superscript 8 baseline plus 2 superscript 7 baseline plus 2 superscript 6 baseline plus 2 superscript 5 baseline plus 2 superscript 4 baseline plus 2 cubed plus 2 squared plus 2 plus 1 equals 2,047 .
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 25+24+23+22+2+1=63
.
is a Mersenne prime. If not, find a factor of 255−1
.(1 point)
Responses
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 222−1
.
No, 2 superscript 55 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 55 baseline minus 1 is 2 superscript 22 baseline minus 1 .
Yes, 255−1
is a Mersenne prime.
Yes, 2 superscript 55 baseline minus 1 is a Mersenne prime.
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 210+29+28+27+26+25+24+23+22+2+1=2,047
.
No, 2 superscript 55 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 55 baseline minus 1 is 2 superscript 10 baseline plus 2 superscript 9 baseline plus 2 superscript 8 baseline plus 2 superscript 7 baseline plus 2 superscript 6 baseline plus 2 superscript 5 baseline plus 2 superscript 4 baseline plus 2 cubed plus 2 squared plus 2 plus 1 equals 2,047 .
No, 255−1
is not a Mersenne prime. A factor of 255−1
is 25+24+23+22+2+1=63
.
Answers
GPT-4o mini
To analyze whether \(2^{55} - 1\) is a Mersenne prime, we first note the definition of a Mersenne prime. A Mersenne prime is a prime number of the form \(2^p - 1\) where \(p\) is a prime number.
In this case, \(p = 55\), which is not a prime number (since it can be factored as \(5 \times 11\)). Therefore, \(2^{55} - 1\) cannot be a Mersenne prime.
Next, to find a factor of \(2^{55} - 1\), we can use the formula for factoring \(2^n - 1\). The factorization states that:
\[
2^{n} - 1 = (2^{d} - 1)(2^{n/d} + 2^{(n/d)-1} + \ldots + 2 + 1)
\]
for any divisor \(d\) of \(n\).
For \(n = 55\), we can look for its divisors. Notably, \(d = 5\) is a divisor. Using it, we can factor:
\[
2^{55} - 1 = (2^{5} - 1)(2^{50} + 2^{45} + \ldots + 1)
\]
Calculating \(2^5 - 1\):
\[
2^5 - 1 = 32 - 1 = 31
\]
Now, let's evaluate this factorization:
- \(2^{5} - 1 = 31\) (which is prime), and
- \(2^{50} + 2^{45} + 2^{40} + 2^{35} + 2^{30} + 2^{25} + 2^{20} + 2^{15} + 2^{10} + 2^{5} + 1\) is a large number that adds together all powers of 2 from \(0\) to \(50\) (in groups).
Since \(2^{55} - 1\) clearly has a factor of \(31\), \(2^{55} - 1\) itself is not prime.
We have established that:
- **No, \(2^{55} - 1\) is not a Mersenne prime.**
- A factor of \(2^{55} - 1\) is \(2^{5} - 1 = 31\).
Now, among the options listed, the correct choice is:
**No, \(2^{55} - 1\) is not a Mersenne prime. A factor of \(2^{55} - 1\) is \(2^{10} + 2^{9} + 2^{8} + 2^{7} + 2^{6} + 2^{5} + 2^{4} + 2^{3} + 2^{2} + 2 + 1 = 2,047.\)**
In this case, \(p = 55\), which is not a prime number (since it can be factored as \(5 \times 11\)). Therefore, \(2^{55} - 1\) cannot be a Mersenne prime.
Next, to find a factor of \(2^{55} - 1\), we can use the formula for factoring \(2^n - 1\). The factorization states that:
\[
2^{n} - 1 = (2^{d} - 1)(2^{n/d} + 2^{(n/d)-1} + \ldots + 2 + 1)
\]
for any divisor \(d\) of \(n\).
For \(n = 55\), we can look for its divisors. Notably, \(d = 5\) is a divisor. Using it, we can factor:
\[
2^{55} - 1 = (2^{5} - 1)(2^{50} + 2^{45} + \ldots + 1)
\]
Calculating \(2^5 - 1\):
\[
2^5 - 1 = 32 - 1 = 31
\]
Now, let's evaluate this factorization:
- \(2^{5} - 1 = 31\) (which is prime), and
- \(2^{50} + 2^{45} + 2^{40} + 2^{35} + 2^{30} + 2^{25} + 2^{20} + 2^{15} + 2^{10} + 2^{5} + 1\) is a large number that adds together all powers of 2 from \(0\) to \(50\) (in groups).
Since \(2^{55} - 1\) clearly has a factor of \(31\), \(2^{55} - 1\) itself is not prime.
We have established that:
- **No, \(2^{55} - 1\) is not a Mersenne prime.**
- A factor of \(2^{55} - 1\) is \(2^{5} - 1 = 31\).
Now, among the options listed, the correct choice is:
**No, \(2^{55} - 1\) is not a Mersenne prime. A factor of \(2^{55} - 1\) is \(2^{10} + 2^{9} + 2^{8} + 2^{7} + 2^{6} + 2^{5} + 2^{4} + 2^{3} + 2^{2} + 2 + 1 = 2,047.\)**
Nanika Zoldyck
Which of the following expressions can be used to determine a Mersenne prime? Enter the number of the correct option.
Option #1: 2n+1
Option #2: 2n−1
Option #3: 2n−1
Option #1: 2n+1
Option #2: 2n−1
Option #3: 2n−1
GPT-4o mini
To determine a Mersenne prime, we specifically refer to the form \(2^p - 1\) where \(p\) is a prime number.
Out of the options provided:
- **Option #1: \(2n + 1\)** does not represent a Mersenne prime since it represents an odd number but is not in the form \(2^p - 1\).
- **Option #2: \(2n - 1\)** is a linear expression and does not relate to Mersenne primes.
- **Option #3: \(2^n - 1\)** is indeed the correct form to check for Mersenne primes, as it directly matches the necessary structure where \(n\) must be a prime number.
Thus, the correct option is **Option #3: \(2^n - 1\)**.
So the answer is **3**.
Out of the options provided:
- **Option #1: \(2n + 1\)** does not represent a Mersenne prime since it represents an odd number but is not in the form \(2^p - 1\).
- **Option #2: \(2n - 1\)** is a linear expression and does not relate to Mersenne primes.
- **Option #3: \(2^n - 1\)** is indeed the correct form to check for Mersenne primes, as it directly matches the necessary structure where \(n\) must be a prime number.
Thus, the correct option is **Option #3: \(2^n - 1\)**.
So the answer is **3**.
Nanika Zoldyck
Which of the following statements must be true about n in the expression for a Mersenne prime, 2n−1
? Enter the number of the correct statement.
Statement #1: It must be an even number.
Statement #2: It must be an odd composite number.
Statement #3: It must be a prime number.
? Enter the number of the correct statement.
Statement #1: It must be an even number.
Statement #2: It must be an odd composite number.
Statement #3: It must be a prime number.
GPT-4o mini
In the context of Mersenne primes, which are of the form \(2^n - 1\), the value of \(n\) must specifically be a prime number for \(2^n - 1\) to potentially be a prime.
Let's analyze the statements:
- **Statement #1: It must be an even number.**
This is not true. The only even prime is 2; however, as a Mersenne prime, \(2^2 - 1 = 3\) is indeed prime, but for any other even number \(n > 2\), \(2^n - 1\) is not prime because \(2^n - 1\) can be factored as \((2^{n/2} - 1)(2^{n/2} + 1)\).
- **Statement #2: It must be an odd composite number.**
This is also not true. For \(n\) to lead to a Mersenne prime, \(n\) must be a prime. An odd composite number (such as 9 or 15) would not satisfy the prime condition.
- **Statement #3: It must be a prime number.**
This statement is true. For \(2^n - 1\) to be potentially prime (i.e., a Mersenne prime), \(n\) must itself be prime.
Thus, the correct statement is **Statement #3: It must be a prime number.**
So the answer is **3**.
Let's analyze the statements:
- **Statement #1: It must be an even number.**
This is not true. The only even prime is 2; however, as a Mersenne prime, \(2^2 - 1 = 3\) is indeed prime, but for any other even number \(n > 2\), \(2^n - 1\) is not prime because \(2^n - 1\) can be factored as \((2^{n/2} - 1)(2^{n/2} + 1)\).
- **Statement #2: It must be an odd composite number.**
This is also not true. For \(n\) to lead to a Mersenne prime, \(n\) must be a prime. An odd composite number (such as 9 or 15) would not satisfy the prime condition.
- **Statement #3: It must be a prime number.**
This statement is true. For \(2^n - 1\) to be potentially prime (i.e., a Mersenne prime), \(n\) must itself be prime.
Thus, the correct statement is **Statement #3: It must be a prime number.**
So the answer is **3**.
Nanika Zoldyck
Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2
to generate a Pythagorean triple when x equals 12 and y equals 5. Which of the following is one of the values of the Pythagorean triple?(1 point)
Responses
119
119
289
289
60
60
49
to generate a Pythagorean triple when x equals 12 and y equals 5. Which of the following is one of the values of the Pythagorean triple?(1 point)
Responses
119
119
289
289
60
60
49
GPT-4o mini
To generate a Pythagorean triple using the polynomial identity
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2,
\]
first, we need to calculate the values given \(x = 12\) and \(y = 5\).
1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 12^2 = 144,
\]
\[
y^2 = 5^2 = 25.
\]
2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 25 = 169.
\]
3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 25 = 119.
\]
4. Calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 5 = 120.
\]
Now, we have determined:
- \(x^2 + y^2 = 169\)
- \(x^2 - y^2 = 119\)
- \(2xy = 120\)
According to the identity, this leads to the equation:
\[
(169)^2 = (119)^2 + (120)^2.
\]
Thus, the Pythagorean triple generated from \(x = 12\) and \(y = 5\) is:
- \(119\) (one leg),
- \(120\) (the other leg),
- \(169\) (the hypotenuse).
Among the options presented (119, 289, 60, and 49), the value of the Pythagorean triple is **119**.
So, the answer is **119**.
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2,
\]
first, we need to calculate the values given \(x = 12\) and \(y = 5\).
1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 12^2 = 144,
\]
\[
y^2 = 5^2 = 25.
\]
2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 25 = 169.
\]
3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 25 = 119.
\]
4. Calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 5 = 120.
\]
Now, we have determined:
- \(x^2 + y^2 = 169\)
- \(x^2 - y^2 = 119\)
- \(2xy = 120\)
According to the identity, this leads to the equation:
\[
(169)^2 = (119)^2 + (120)^2.
\]
Thus, the Pythagorean triple generated from \(x = 12\) and \(y = 5\) is:
- \(119\) (one leg),
- \(120\) (the other leg),
- \(169\) (the hypotenuse).
Among the options presented (119, 289, 60, and 49), the value of the Pythagorean triple is **119**.
So, the answer is **119**.
Nanika Zoldyck
Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2
to generate a Pythagorean triple when x equals 7 and y equals 4. Write the Pythagorean triple in ascending order.(2 points)
to generate a Pythagorean triple when x equals 7 and y equals 4. Write the Pythagorean triple in ascending order.(2 points)
GPT-4o mini
To generate a Pythagorean triple using the polynomial identity
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2,
\]
we can plug in the values \(x = 7\) and \(y = 4\).
1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 7^2 = 49,
\]
\[
y^2 = 4^2 = 16.
\]
2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 49 + 16 = 65.
\]
3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 49 - 16 = 33.
\]
4. Calculate \(2xy\):
\[
2xy = 2 \cdot 7 \cdot 4 = 56.
\]
Now we have three values:
- \(x^2 + y^2 = 65\) (this corresponds to the hypotenuse).
- \(x^2 - y^2 = 33\) (this corresponds to one leg).
- \(2xy = 56\) (this corresponds to the other leg).
So, the Pythagorean triple formed from \(x = 7\) and \(y = 4\) is:
- \(33\),
- \(56\),
- \(65\).
In ascending order, the Pythagorean triple is:
\[
(33, 56, 65).
\]
Thus, the final answer is **33, 56, 65**.
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2,
\]
we can plug in the values \(x = 7\) and \(y = 4\).
1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 7^2 = 49,
\]
\[
y^2 = 4^2 = 16.
\]
2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 49 + 16 = 65.
\]
3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 49 - 16 = 33.
\]
4. Calculate \(2xy\):
\[
2xy = 2 \cdot 7 \cdot 4 = 56.
\]
Now we have three values:
- \(x^2 + y^2 = 65\) (this corresponds to the hypotenuse).
- \(x^2 - y^2 = 33\) (this corresponds to one leg).
- \(2xy = 56\) (this corresponds to the other leg).
So, the Pythagorean triple formed from \(x = 7\) and \(y = 4\) is:
- \(33\),
- \(56\),
- \(65\).
In ascending order, the Pythagorean triple is:
\[
(33, 56, 65).
\]
Thus, the final answer is **33, 56, 65**.