Express each of the following numbers as the sum of three odd primes:
31

Answer

To express the number 31 as the sum of three odd prime numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Odd Prime Numbers Less Than 31**:
The odd prime numbers less than 31 are:
- 3, 5, 7, 11, 13, 17, 19, 23, 29.

2. **Set Up the Equation**:
We need to find three odd primes \( p_1, p_2, p_3 \) such that:
\[ p_1 + p_2 + p_3 = 31 \]

3. **Start Testing Combinations**:
We can start by testing combinations of these odd primes to see which ones add up to 31.

- **Combination 1**:
Let's try \( 3 + 5 + 23 \):
\[ 3 + 5 + 23 = 31 \]
This works!

- **Combination 2**:
Let's try \( 7 + 11 + 13 \):
\[ 7 + 11 + 13 = 31 \]
This also works!

4. **List the Results**:
We have found two valid combinations:
- \( 3 + 5 + 23 = 31 \)
- \( 7 + 11 + 13 = 31 \)

5. **Conclusion**:
Therefore, we can express 31 as the sum of three odd primes in multiple ways:
- \( 31 = 3 + 5 + 23 \)
- \( 31 = 7 + 11 + 13 \)