Let $\text{Prime}(x)$ be the $x$th prime, so $\text{Prime}(1) = 2$. My question is regarding sums modulo the product of the first $n$ primes, i.e. $\bmod Q$ with $Q = \prod_{k=1}^n{\text{Prime}(k)}$.
How many primes $p_i$ with $\text{Prime}(n) < p_i < Q$ do we need to sum to any number less than $Q$, modulo $Q$?.
EXAMPLES
For example, modulo $Q = 2 \cdot 3 \cdot 5$, if we take $n=2$ primes, we can get $14$ by adding $p_1=7$ plus $p_2=7$, i.e. $7+7=14$. We can get $18$ by adding $7+11$. We can get $28$ by adding $29+29 \bmod 2 \cdot 3 \cdot 5 = 28$.
I'm wondering how many primes we need to be able to sum to any number modulo $Q$. I conjecture that we may only need up to 3 primes, to be able to use 1, 2, or 3 primes to sum to a given number.