To determine if the square of a prime number is also a prime, we need to understand the properties of prime numbers and squares.
### Step-by-Step Analysis:
1. Identify a Prime Number:
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11, etc.
2. Square the Prime Number:
Squaring a prime number means multiplying the prime number by itself. For example, if we take the prime number [tex]\( p \)[/tex], the square is [tex]\( p^2 \)[/tex].
3. Check if the Square is Prime:
To check if the square of a given prime number is also prime, we need to determine if the squared number has any divisors other than 1 and itself.
### Example Illustrations:
- Prime Number 2:
- Square: [tex]\( 2^2 = 4 \)[/tex]
- Divisors of 4: 1, 2, 4
- Since 4 has divisors other than 1 and 4 (specifically, 2), it is not a prime number.
- Prime Number 3:
- Square: [tex]\( 3^2 = 9 \)[/tex]
- Divisors of 9: 1, 3, 9
- Since 9 has divisors other than 1 and 9 (specifically, 3), it is not a prime number.
- Prime Number 5:
- Square: [tex]\( 5^2 = 25 \)[/tex]
- Divisors of 25: 1, 5, 25
- Since 25 has divisors other than 1 and 25 (specifically, 5), it is not a prime number.
### General Conclusion:
We can see that when we square a prime number [tex]\( p \)[/tex], the result [tex]\( p^2 \)[/tex] is always divisible by [tex]\( p \)[/tex]. This means [tex]\( p^2 \)[/tex] has at least three divisors: 1, [tex]\( p \)[/tex], and [tex]\( p^2 \)[/tex]. Therefore, [tex]\( p^2 \)[/tex] cannot be prime because a prime number only has exactly two distinct positive divisors, 1 and the number itself.
### Final Answer:
The square of a prime number is not a prime number.
Thus, the answer to the question "Is the square of a prime number also prime?" is:
False.
