Certainly! Let's tackle your questions one by one with clear explanations.
### 1. List all prime numbers between 1 and 50
A prime number is a natural number greater than 1 that is not divisible by any positive integers other than 1 and itself. Here's the list:
- 2
- 3
- 5
- 7
- 11
- 13
- 17
- 19
- 23
- 29
- 31
- 37
- 41
- 43
- 47
So, the prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
### 2. Determine if the given numbers are prime or not
Now, let's analyze the given numbers:
#### 26
- Explanation: 26 is not a prime number because it can be factored into [tex]\( 13 \times 2 \)[/tex].
#### 81
- Explanation: 81 is not a prime number because it can be factored into [tex]\( 3 \times 3 \times 3 \times 3 \)[/tex] (or [tex]\( 3^4 \)[/tex]).
#### 27
- Explanation: 27 is not a prime number because it can be factored into [tex]\( 3 \times 3 \times 3 \)[/tex] (or [tex]\( 3^3 \)[/tex]).
#### 83
- Explanation: 83 is a prime number because it cannot be divided evenly by any number other than 1 and itself.
#### 51
- Explanation: 51 is not a prime number because it can be factored into [tex]\( 17 \times 3 \)[/tex].
#### 99
- Explanation: 99 is not a prime number because it can be factored into [tex]\( 3 \times 3 \times 11 \)[/tex] (or [tex]\( 3^2 \times 11 \)[/tex]).
#### 57
- Explanation: 57 is not a prime number because it can be factored into [tex]\( 3 \times 19 \)[/tex].
#### 101
- Explanation: 101 is a prime number because it cannot be divided evenly by any number other than 1 and itself.
### Summary Table
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number} & \text{Prime Status} \\
\hline
26 & \text{No} \\
81 & \text{No} \\
27 & \text{No} \\
83 & \text{Yes} \\
51 & \text{No} \\
99 & \text{No} \\
57 & \text{No} \\
101 & \text{Yes} \\
\hline
\end{array}
\][/tex]
In summary, the primes between 1 and 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Among the given numbers, 83 and 101 are prime, while 26, 81, 27, 51, 99, and 57 are not.
