Which of the following numbers is a prime in the form 2n−1?(1 point)
1. 127
2. 63
3. 15
4. 33
The numbers in the form 2n-1 are prime if and only if n is a positive integer.
1. 127: This number is in the form 2n-1 where n = 64. It is a prime number.
2. 63: This number is not in the form 2n-1.
3. 15: This number is not in the form 2n-1.
4. 33: This number is not in the form 2n-1.
Therefore, the prime number in the form 2n-1 is 127.
Answer: 1. 127
To determine which of the given numbers is a prime in the form 2n−1, we need to check if each number satisfies the conditions for primality.
1. 127: To check if it is prime, we can try dividing it by prime numbers less than its square root, which is approximately 11. The closest prime number to 127 less than 11 is 7. Dividing 127 by 7 gives us a remainder of 2. Since there is no whole number divisor less than 11, we can conclude that 127 is a prime number in the form 2n−1.
2. 63: To determine if it is prime, we can try dividing it by prime numbers less than its square root, which is approximately 8. The closest prime number to 63 less than 8 is 7. Dividing 63 by 7 gives us a remainder of 0. Since there is a whole number divisor less than 8, we can conclude that 63 is not a prime number in the form 2n−1.
3. 15: To check if it is prime, we can try dividing it by prime numbers less than its square root, which is approximately 4. The closest prime number to 15 less than 4 is 3. Dividing 15 by 3 gives us a remainder of 0. Since there is a whole number divisor less than 4, we can conclude that 15 is not a prime number in the form 2n−1.
4. 33: To determine if it is prime, we can try dividing it by prime numbers less than its square root, which is approximately 6. The closest prime number to 33 less than 6 is 5. Dividing 33 by 5 gives us a remainder of 3. Since there is no whole number divisor less than 6, we can conclude that 33 is a prime number in the form 2n−1.
Based on the above analysis, the numbers that are prime in the form 2n−1 are:
- 127 and 33
Therefore, the correct answer is option 1. 127 and option 4. 33.
To determine which of the given numbers is a prime in the form 2n−1, we need to check if any of the numbers satisfy the conditions.
The form 2n−1 is equivalent to the expression 2^n - 1. In other words, we raise 2 to some power n, and then subtract 1 from the result.
Let's go through each option:
1. 127: We check if it can be expressed in the form 2^n - 1. If we try different values of n, we find that 2^7 - 1 = 128 - 1 = 127. So 127 is in the form 2n−1.
2. 63: We try to express it in the given form. However, for any value of n we try, the result will always be even, since 2^n will be a power of 2 (which is always even) minus 1. Therefore, 63 is not in the form 2n−1.
3. 15: Similar to the previous option, when we try to express it in the given form, we find that 2^n will always be even, and subtracting 1 from an even number will always result in an odd number. Therefore, 15 is not in the form 2n−1.
4. 33: Again, we can use the same reasoning as before. No matter the value of n, the expression 2^n - 1 will always result in an even number, which means 33 is not in the form 2n−1.
Therefore, the only number from the given options that is a prime in the form 2n−1 is 127, option 1.