TITLE: Can the residues modulo small integers uniquely determine a number?
QUESTION [5 upvotes]: I was on Wolfram Alpha  looking at the factors of a number, and I saw the section titled "Residues modulo small integers". It got me wondering whether the residues for $2, \ldots, 9$ would uniquely determine a number, or possibly provide a equivalence class. If not, how many residues would you need? (And would they have to be be modulo prime numbers?)

REPLY [5 votes]: Systems of linear congruences are the subject of various forms of the chinese remainder theorem. In its original form it is about pairwise co-prime remainders.
Here we have
$$
x \equiv r_k \pmod k \quad (k \in \{2,\dots,9\})
$$
If a solution $x$ exists, then all solutions are 
$$
\{ x + k M \mid k \in \mathbb{Z} \}, \quad M := \mbox{lcm}(r_k) 
$$
So there is no uniqueness. An alternative would be to ask for the smallest positive integer solution.
A criterion for the existence of a solution is
$$
r_i \equiv r_j \pmod{ \gcd(i,j)}
$$
See theorem 2 of the above link.