To find the mystery number, we will first list out the prime numbers between 50 and 100:
53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Now, let's go through each of these numbers and see if we can make exactly 5 different rectangles with a mystery number of tiles using them:
1. For 53:
- We can have a rectangle with dimensions 1 x 53 (using 53 tiles).
- We can have a rectangle with dimensions 53 x 1 (using 53 tiles).
- We cannot make any additional rectangles with the given condition.
2. For 59:
- We can have a rectangle with dimensions 1 x 59 (using 59 tiles).
- We can have a rectangle with dimensions 59 x 1 (using 59 tiles).
- We cannot make any additional rectangles with the given condition.
3. For 61:
- We can have a rectangle with dimensions 1 x 61 (using 61 tiles).
- We can have a rectangle with dimensions 61 x 1 (using 61 tiles).
- We cannot make any additional rectangles with the given condition.
4. For 67:
- We can have a rectangle with dimensions 1 x 67 (using 67 tiles).
- We can have a rectangle with dimensions 67 x 1 (using 67 tiles).
- We cannot make any additional rectangles with the given condition.
5. For 71:
- We can have a rectangle with dimensions 1 x 71 (using 71 tiles).
- We can have a rectangle with dimensions 71 x 1 (using 71 tiles).
- We cannot make any additional rectangles with the given condition.
6. For 73:
- We can have a rectangle with dimensions 1 x 73 (using 73 tiles).
- We can have a rectangle with dimensions 73 x 1 (using 73 tiles).
- We cannot make any additional rectangles with the given condition.
7. For 79:
- We can have a rectangle with dimensions 1 x 79 (using 79 tiles).
- We can have a rectangle with dimensions 79 x 1 (using 79 tiles).
- We cannot make any additional rectangles with the given condition.
8. For 83:
- We can have a rectangle with dimensions 1 x 83 (using 83 tiles).
- We can have a rectangle with dimensions 83 x 1 (using 83 tiles).
- We cannot make any additional rectangles with the given condition.
9. For 89:
- We can have a rectangle with dimensions 1 x 89 (using 89 tiles).
- We can have a rectangle with dimensions 89 x 1 (using 89 tiles).
- We cannot make any additional rectangles with the given condition.
10. For 97:
- We can have a rectangle with dimensions 1 x 97 (using 97 tiles).
- We can have a rectangle with dimensions 97 x 1 (using 97 tiles).
- We cannot make any additional rectangles with the given condition.
After going through all the numbers, we can see that none of them satisfy the given condition of making exactly 5 different rectangles. It is possible that there is no number that meets all the criteria given.