Jo needs to know which of the following cannot be a prime number whenever p is a prime number greater than 2001.
A. p-2
B. p+2
C. p+999
D. p+9990
Im stuck
if p>2 is prime, p is odd
So, p+999 is even, so it cannot be prime.
oh ok thanks!
To determine which of the options cannot be a prime number, we need to consider the properties of prime numbers.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number cannot be evenly divided by any other number apart from 1 and itself.
Let's evaluate each option in the given choices:
A. p-2:
Here, we subtract 2 from the prime number. Since p is a prime number greater than 2001, p-2 will always be a number smaller than p. Hence, it has a higher chance of being a prime number. However, we cannot guarantee that it is always a prime number. For example, if p = 2003, then p-2 = 2001, which is not a prime number. Therefore, option A can be a prime number sometimes.
B. p+2:
In this case, we add 2 to the prime number. Since p is a prime number greater than 2001, p+2 will always be a number greater than p. Again, it has the potential to be a prime number, but we cannot guarantee it. For example, if p = 2003, then p+2 = 2005, which is a prime number. Therefore, option B can be a prime number sometimes.
C. p+999:
Here, we add 999 to the prime number. Since p is a prime number greater than 2001, p+999 will always be a number greater than p. However, 999 is not a prime number itself, as it can be divided by 3 and 37. Hence, p+999 will also be divisible by 3 and 37, making it not a prime number. Therefore, option C cannot be a prime number.
D. p+9990:
In this scenario, we add 9990 to the prime number. As mentioned earlier, p is a prime number greater than 2001, so p+9990 will always be a number greater than p. Like option C, 9990 is also not a prime number since it can be divided by 2, 3, 5, 6, 10, 15, 30, 71, 142, 213, 355, 426, 710, 1065, 2130, and 9990. Therefore, adding a non-prime number (9990) to a prime number will result in a number that is not a prime. Hence, option D cannot be a prime number.
To summarize:
- Option A (p-2) can be a prime number sometimes.
- Option B (p+2) can be a prime number sometimes.
- Option C (p+999) cannot be a prime number.
- Option D (p+9990) cannot be a prime number.
Therefore, the answer is C. p+999 cannot be a prime number whenever p is a prime number greater than 2001.