To find the sum of all the prime numbers between 10 and 30, we first need to identify which of the numbers in this range are prime. Let's recall that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
Here are the step-by-step details:
1. List the numbers between 10 and 30:
- These numbers are: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30.
2. Identify the prime numbers from this list:
- 10 is not a prime number (divisible by 2 and 5).
- 11 is a prime number (divisible only by 1 and 11).
- 12 is not a prime number (divisible by 2, 3, 4, and 6).
- 13 is a prime number (divisible only by 1 and 13).
- 14 is not a prime number (divisible by 2 and 7).
- 15 is not a prime number (divisible by 3 and 5).
- 16 is not a prime number (divisible by 2, 4, and 8).
- 17 is a prime number (divisible only by 1 and 17).
- 18 is not a prime number (divisible by 2, 3, 6, and 9).
- 19 is a prime number (divisible only by 1 and 19).
- 20 is not a prime number (divisible by 2, 4, 5, and 10).
- 21 is not a prime number (divisible by 3 and 7).
- 22 is not a prime number (divisible by 2 and 11).
- 23 is a prime number (divisible only by 1 and 23).
- 24 is not a prime number (divisible by 2, 3, 4, 6, 8, and 12).
- 25 is not a prime number (divisible by 5).
- 26 is not a prime number (divisible by 2 and 13).
- 27 is not a prime number (divisible by 3 and 9).
- 28 is not a prime number (divisible by 2, 4, 7, and 14).
- 29 is a prime number (divisible only by 1 and 29).
- 30 is not a prime number (divisible by 2, 3, 5, 6, 10, and 15).
The prime numbers between 10 and 30 are: 11, 13, 17, 19, 23, and 29.
3. Sum the identified prime numbers:
- Adding these prime numbers together: [tex]\(11 + 13 + 17 + 19 + 23 + 29\)[/tex].
4. Calculate the sum:
- [tex]\(11 + 13 = 24\)[/tex]
- [tex]\(24 + 17 = 41\)[/tex]
- [tex]\(41 + 19 = 60\)[/tex]
- [tex]\(60 + 23 = 83\)[/tex]
- [tex]\(83 + 29 = 112\)[/tex]
So, the sum of all the prime numbers between 10 and 30 is [tex]\( \boxed{112} \)[/tex].
