In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:


For every $n$ elements $v\_1, \ldots, v\_n$ that generate the unit ideal there are numbers $r\_1, \ldots r\_{n-1}$ such that $v\_1 + r\_1 v\_n, v\_2 + r\_2 v\_n, \ldots, v\_{n-1} + r\_{n-1} v\_n$ also generate the unit ideal. 


He then goes on to show that a noetherian, d-dimensional ring has this property for all $n \geq d+2$, but the proof is long and nontrivial.


My question now is: Is there an easier way to see this for a principal ideal domain and say $n=3$? 
Or even more concretely, given three numbers $a,b,c \in \mathbb Z$ with $gcd(a,b,c) = 1$ why are there numbers $n,m \in \mathbb Z$, such that also $gcd(a+nc,b+mc) = 1$?


P.S. I have a more technical question along the same lines waiting for the lucky answerer! All of this comes from my trying to understand van-der-Kallen homology stability of general linear groups.



Bass' stable range condition for principal ideal domains
EDIT 3 : Sorry for editing this old answer one more time, but I want to also point out for future readers that there is a proof that Dedekind domains have stable range $2$ which is very similar to my proof for PID's in Satz K.13 of the book *Algebra* by Jantzen and Schwermer. The whole Appendix K of that book is a lovely introduction to the whole notion of stable range.




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EDIT 2 : I just learned of a super-short proof of the special case of the Bass Stable Range theorem alluded to in the question (the one giving the stable range for Noetherian $d$-dimensional rings). It's a little more abstract than what I did below for PID's, but not much harder. See Section 2 of 


MR0217052 (36 #147)
Estes, Dennis; Ohm, Jack
Stable range in commutative rings. 
J. Algebra 7 1967 343–362. 




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EDIT : Here's a proof that works for $R$ a a PID, which implies that the condition of generating the unit ideal is the same as having gcd equal to $1$.


For some $n \geq 2$ consider a tuple $(a\_1,\ldots,a\_{n+1})$ of elements of $R$ whose gcd is $1$. We want to find $r\_1,\ldots,r\_n \in R$ such that $\text{gcd}(a\_1+r\_1 a\_{n+1},\ldots,a\_n + r\_n a\_{n+1}) = 1$. 


There are three cases. If $a\_{n+1}=0$, then there is nothing to do. If $a\_i=0$ for some $1 \leq i \leq n$, then we can take $r\_i=1$ and $r\_j=0$ for $j \neq i$.


The most interesting case is when none of the $a\_i$ equal $0$. In this case, we will only need $r\_1$ (the rest of the $r\_i$ can be taken to be $0$). Set $b = \text{gcd}(a\_2,\ldots,a\_n)$,
and let $p\_1,\ldots,p\_k$ be the distinct primes dividing $b$. For each $i$, we know that $p\_i$ cannot divide both $a\_1$ and $a\_{n+1}$. This implies that there exists some $c\_i \in \{0,1\}$ such that
$$a\_1 + c\_i a\_{n+1} \neq 0 \quad (\text{mod } p\_i).$$
By the Chinese remainder theorem, there exists some $r\_1 \in R$ such that
$$r\_1 = c\_i \quad (\text{mod } p\_i)$$
for $1 \leq i \leq k$, which implies that 
$$a\_1 + r\_1 a\_{n+1} \neq 0 \quad (\text{mod } p\_i)$$
for all $1 \leq i \leq k$. We conclude that the gcd of $a\_1+r\_1 a\_{n+1}$ and $b$ equals $1$, and thus that the gcd of $a\_1+r\_1 a\_{n+1},a\_2,\ldots,a\_n$ is $1$.




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Here is what was my original answer:


This does not exactly answer your question, but it is much easier to prove that the complexes that van der Kallen needs are highly connected for $\mathbb{Z}$ than for general rings. This was originally done by Maazen in his unpublished thesis, which can be downloaded [here](http://dspace.library.uu.nl/handle/1874/237657). There is also a different proof of this connectivity in Step 2 of the proof of Theorem B in my paper "The complex of partial bases for $F\_n$ and finite generation of the Torelli subgroup of $\text{Aut}(F\_n)$" with Matt Day, available on my [webpage](http://www.nd.edu/~andyp/papers/).


