rogue wrote: 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? No. $31$ can not.

However, it is true for positive integers $n$ larger than some value. It is trivial for composite numbers $n$ larger than $9$. It is also true for large enough primes $n=p$. Then $m = p-21$ is even, and any even number $m \geq 32$ can be represented as $m=6a+10b$ for some positive integers $a,b$. But then $p = 6a+10b+6+15$. Thus the claim is true for all numbers $n>47$. But $47 = 6+6+15+20$, $43=6+10+12+15$, $41=6+10+10+15$, $37 = 6+6+10+15$, so in fact the claim is true for all numbers $\boxed{n>31}$.