The number of positive integers 'n' for which `3n-4, 4n-5` and 5n - 3 are all primes is:

Answer

To solve the problem of finding the number of positive integers \( n \) for which \( 3n-4 \), \( 4n-5 \), and \( 5n-3 \) are all prime numbers, we will evaluate these expressions for various values of \( n \).

### Step-by-Step Solution:

1. **Evaluate the expressions for small positive integers \( n \)**:
- We will start by substituting values of \( n \) from 1 upwards and check if all three expressions yield prime numbers.

2. **Check \( n = 1 \)**:
- \( 3n - 4 = 3(1) - 4 = -1 \) (not prime)
- \( 4n - 5 = 4(1) - 5 = -1 \) (not prime)
- \( 5n - 3 = 5(1) - 3 = 2 \) (prime)
- **Conclusion**: Not valid since two expressions are not prime.

3. **Check \( n = 2 \)**:
- \( 3n - 4 = 3(2) - 4 = 2 \) (prime)
- \( 4n - 5 = 4(2) - 5 = 3 \) (prime)
- \( 5n - 3 = 5(2) - 3 = 7 \) (prime)
- **Conclusion**: All three expressions are prime.

4. **Check \( n = 3 \)**:
- \( 3n - 4 = 3(3) - 4 = 5 \) (prime)
- \( 4n - 5 = 4(3) - 5 = 7 \) (prime)
- \( 5n - 3 = 5(3) - 3 = 12 \) (not prime)
- **Conclusion**: Not valid since one expression is not prime.

5. **Check \( n = 4 \)**:
- \( 3n - 4 = 3(4) - 4 = 8 \) (not prime)
- \( 4n - 5 = 4(4) - 5 = 11 \) (prime)
- \( 5n - 3 = 5(4) - 3 = 17 \) (prime)
- **Conclusion**: Not valid since one expression is not prime.

6. **Check \( n = 5 \)**:
- \( 3n - 4 = 3(5) - 4 = 11 \) (prime)
- \( 4n - 5 = 4(5) - 5 = 15 \) (not prime)
- \( 5n - 3 = 5(5) - 3 = 22 \) (not prime)
- **Conclusion**: Not valid since two expressions are not prime.

7. **Check \( n = 6 \)**:
- \( 3n - 4 = 3(6) - 4 = 14 \) (not prime)
- \( 4n - 5 = 4(6) - 5 = 19 \) (prime)
- \( 5n - 3 = 5(6) - 3 = 27 \) (not prime)
- **Conclusion**: Not valid since two expressions are not prime.

8. **Check \( n = 7 \)**:
- \( 3n - 4 = 3(7) - 4 = 18 \) (not prime)
- \( 4n - 5 = 4(7) - 5 = 23 \) (prime)
- \( 5n - 3 = 5(7) - 3 = 32 \) (not prime)
- **Conclusion**: Not valid since two expressions are not prime.

9. **Check \( n = 8 \)**:
- \( 3n - 4 = 3(8) - 4 = 20 \) (not prime)
- \( 4n - 5 = 4(8) - 5 = 27 \) (not prime)
- \( 5n - 3 = 5(8) - 3 = 37 \) (prime)
- **Conclusion**: Not valid since two expressions are not prime.

10. **Check \( n = 9 \)**:
- \( 3n - 4 = 3(9) - 4 = 23 \) (prime)
- \( 4n - 5 = 4(9) - 5 = 31 \) (prime)
- \( 5n - 3 = 5(9) - 3 = 42 \) (not prime)
- **Conclusion**: Not valid since one expression is not prime.

### Summary:
After checking all values from \( n = 1 \) to \( n = 9 \), the only positive integer \( n \) for which \( 3n-4 \), \( 4n-5 \), and \( 5n-3 \) are all prime is \( n = 2 \).

Thus, the number of positive integers \( n \) for which all three expressions are prime is **1**.