Alright, let's identify the prime numbers from the given lists.
### What is a Prime Number?
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A prime number has exactly two distinct positive divisors: 1 and itself.
### Let's Examine the Given Lists
#### First List: [tex]\(101, 103, 105, 107, 109, 111\)[/tex]
1. 101:
- Divisors: Only divisible by 1 and 101.
- Prime
2. 103:
- Divisors: Only divisible by 1 and 103.
- Prime
3. 105:
- Check divisibility: 105 is divisible by 3 and 5 (105 = 3 5 7).
- Not a Prime
4. 107:
- Divisors: Only divisible by 1 and 107.
- Prime
5. 109:
- Divisors: Only divisible by 1 and 109.
- Prime
6. 111:
- Check divisibility: 111 is divisible by 3 (111 = 3 37).
- Not a Prime
From this list, the prime numbers are: [tex]\(101, 103, 107, 109\)[/tex].
#### Second List: [tex]\(123, 137, 145, 177, 189, 801\)[/tex]
1. 123:
- Check divisibility: 123 is divisible by 3 (123 = 3 41).
- Not a Prime
2. 137:
- Divisors: Only divisible by 1 and 137.
- Prime
3. 145:
- Check divisibility: 145 is divisible by 5 (145 = 5 29).
- Not a Prime
4. 177:
- Check divisibility: 177 is divisible by 3 (177 = 3 59).
- Not a Prime
5. 189:
- Check divisibility: 189 is divisible by 3 (189 = 3 63).
- Not a Prime
6. 801:
- Check divisibility: 801 is divisible by 3 (801 = 3 267).
- Not a Prime
From this list, the only prime number is: [tex]\(137\)[/tex].
### Final Results
After checking each number in the given lists, the prime numbers are:
- From the first list: [tex]\(101, 103, 107, 109\)[/tex]
- From the second list: [tex]\(137\)[/tex]
So, the prime numbers you should circle are:
First List:
[tex]\[101, 103, 107, 109\][/tex]
Second List:
[tex]\[137\][/tex]
