The sum of three numbers is 100, two are prime and one is the sum of the other two.

(a) What is the largest of the three numbers?
(b) Give an example of such three numbers.

(c) How many solutions exist for this problem?
(a) Initially observe that, since the sum of the three numbers is 100 and the largest of them is equal to the sum of the other two, then twice the largest number is 100, that is, the largest number is 50.

(b) Since 50 is not a prime number, the other two numbers are primes and their sum is 50. For example, 3 and 47 are primes and $3+47=50$. Therefore, the numbers 3, 47, and 50 form a solution to the problem.

(c) There are other solutions to the problem. To find them, we write down the list of all primes between 1 and 50, that is, $2,3,5,7,11,13,17,19,23,29,31,37,41,43$ and 47, and for each of these numbers, we check if the difference from 50 is also a prime. We find a total of four solutions.

| Solution 1 | 3 | 47 | 50 |
| :--- | :---: | :---: | :---: |
| Solution 2 | 7 | 43 | 50 |
| Solution 3 | 13 | 37 | 50 |
| Solution 4 | 19 | 31 | 50 |