Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?

To determine if every positive integer greater than 30 can be expressed as the sum of four positive integers such that each pair of these integers has a common divisor greater than 1, we will proceed with the following steps:

1. **Consider Composite Numbers Greater than 9:**
   - For composite numbers \( n \) larger than 9, it is trivial to find such a representation. For example, if \( n \) is composite, it can be written as a sum of four integers where each pair shares a common divisor greater than 1.

2. **Consider Large Prime Numbers \( n = p \):**
   - For large enough primes \( n = p \), we can use the fact that \( m = p - 21 \) is even. Any even number \( m \geq 32 \) can be represented as \( m = 6a + 10b \) for some positive integers \( a \) and \( b \).

3. **Express \( p \) in Terms of \( a \) and \( b \):**
   - Given \( m = 6a + 10b \), we have \( p = 6a + 10b + 6 + 15 \). This shows that \( p \) can be written as a sum of four integers \( 6, 10, 15, \) and another integer \( 6a + 10b \).

4. **Verify for Specific Values:**
   - We need to check if the claim holds for all numbers \( n > 31 \). We verify this by checking specific values:
     - \( 47 = 6 + 6 + 15 + 20 \)
     - \( 43 = 6 + 10 + 12 + 15 \)
     - \( 41 = 6 + 10 + 10 + 15 \)
     - \( 37 = 6 + 6 + 10 + 15 \)

5. **Conclusion:**
   - Since we have verified that the claim holds for numbers 37, 41, 43, and 47, and since the claim is trivial for composite numbers larger than 9 and large enough primes, we conclude that the claim is true for all numbers \( n > 31 \).

The final answer is \( \boxed{ n > 31 } \)