TITLE: Adding coprimes to get a coprime
QUESTION [6 upvotes]: These questions arose in my research. Let $n$ be odd and let  $\mathcal{C}_n$ be the set of integers less than $n$ which are coprime with $n$. 
Question 1: For each integer $\ell: 0 \leq \ell < n$ can we always find a subset $\mathcal{S} \subset \mathcal{C}_n$ such that
$$ \ell+2\cdot\sum_{c \in \mathcal{S}}c \quad (\textrm{mod } n)$$
is coprime with n?
That is, can we always find some set of numbers coprime with $n$ such that twice their sum added to $\ell$ is coprime with $n$? Can we always get $|\mathcal{S}| = 1$ when $\ell$ is not coprime with $n$?
Question 2: What is the smallest set $\mathcal{B_n} \subset \mathcal{C}_n$ such that Question 1 holds with $\mathcal{C}_n$ replaced by $\mathcal{B}_n$?

REPLY [2 votes]: To make up for my  not-well-thought-out comment to Fedor Petrov's answer, here is an answer to show $B$ can have $O((\log k)^2)$ elements, where $k$ is the number of distinct prime factors of $n$.  If one pursues the literature, one can take this down to $O(\log k)$ elements.
$n$ being odd, $2$ and its powers are coprime to $n$, with $d=(n+1)/2$ being a multiplicative inverse mod $n$.  Now $\gcd(l+2b,n)=\gcd(dl+2db,n)=\gcd( c+b,n)$, so we just need to find the greatest distance between an arbitrary number $c$ and the next largest coprime integer $c+b$.  More simply put, $l+2i$ is coprime to $n$ for at least one $i$ greater than $0$ and less than a value which I will call $g(n)+1$.
This is related to Jacobsthal's function $g(n)$, of which I know a little something.  One has $g(n) \leq k^{A + C\log\log k}$, where $A,C \lt 4$, which can be found in an arxiv preprint (cf MathOverflow 37679), which lets us take $B$ to be the powers of 2 from 1 up to $\log(g(n))$, giving fewer than $O((\log k)^2)$ elements.
Gerhard "Sorry For Not Thinking Earlier" Paseman, 2016.01.13