The sum of 3 numbers is 100, two are prime and one is the sum of the other two.

(a) What is the largest of the 3 numbers?

(b) Give an example of these 3 numbers.

(c) How many solutions exist for this problem?
(a) Initially observe that:

- the largest number is the sum of the other two;
- the largest number cannot exceed 50, otherwise the sum of the three would be greater than 100;
- the largest number cannot be less than 50, otherwise the sum of the three would be less than 100.

Therefore, the largest number can only be 50.

(b) The numbers 3, 47, and 50 form a solution to the problem.

(c) There are as many solutions as there are pairs of primes that sum to 50. The table shows all the solutions. Therefore, this problem has 4 solutions.

| 3 | 47 | 50 |
| :---: | :---: | :---: |
| 7 | 43 | 50 |
| 13 | 37 | 50 |
| 19 | 31 | 50 |