Express each of the following numbers as the sum of three odd primes:
21

Answer

To express the number 21 as the sum of three odd prime numbers, we can follow these steps:

### Step 1: Identify Odd Prime Numbers Less Than 21
First, we need to list all the odd prime numbers that are less than 21. The odd prime numbers are:
- 3
- 5
- 7
- 11
- 13
- 17
- 19

### Step 2: Find Combinations of Three Odd Primes
Next, we need to find combinations of these odd prime numbers that add up to 21. We can try different combinations:

1. **Combination 1**: 3 + 5 + 13
- 3 + 5 = 8
- 8 + 13 = 21 (Valid combination)

2. **Combination 2**: 3 + 7 + 11
- 3 + 7 = 10
- 10 + 11 = 21 (Valid combination)

3. **Combination 3**: 5 + 5 + 11
- 5 + 5 = 10
- 10 + 11 = 21 (Valid combination)

4. **Combination 4**: 3 + 3 + 15 (15 is not prime)
- This combination does not work as 15 is not an odd prime.

5. **Combination 5**: 7 + 7 + 7
- 7 + 7 + 7 = 21 (Valid combination)

### Step 3: Write the Final Answer
From our combinations, we found several valid ways to express 21 as the sum of three odd primes:
- 3 + 5 + 13
- 3 + 7 + 11
- 7 + 7 + 7

Thus, we can express 21 as:
- **21 = 3 + 5 + 13**
- **21 = 3 + 7 + 11**
- **21 = 7 + 7 + 7**