To express a number as the sum of three odd primes, we start by identifying odd prime numbers that can be combined to reach the target number. Below, we break down how to achieve this for each of the specified numbers:
1. Expressing 21 as the sum of three odd primes:
- 7 + 7 + 7 = 21
- 3 + 5 + 13 = 21
- 5 + 5 + 11 = 21
Thus, there are several combinations for 21 involving three odd primes.
2. Expressing 31 as the sum of three odd primes:
- 3 + 11 + 17 = 31
- 5 + 13 + 13 = 31
- 7 + 11 + 13 = 31
Various combinations allow us to achieve the total of 31 using odd primes.
3. Expressing 53 as the sum of three odd primes:
- 3 + 7 + 43 = 53
- 3 + 19 + 31 = 53
- 5 + 23 + 25 = 53
Again, multiple combinations can add up to 53 while adhering to the criteria of being odd primes.
4. Expressing 61 as the sum of three odd primes:
- 3 + 29 + 29 = 61
- 7 + 23 + 31 = 61
- 5 + 25 + 31 = 61
We also find that 61 can be reflected in different ways through the summation of odd primes.
In conclusion, each number can be expressed as the sum of three odd primes, and a variety of combinations exist for efficient representations.