What's the smallest three digit number that satisfies the following system of congruences? So my question is how to find the smallest three digit number $n$, which satisfies the following:
$$
x \equiv 0\mod2\\ 
x \equiv 2\mod3\\
x \equiv 2\mod4\\
x \equiv 2\mod5\\
x \equiv 2\mod6\\
$$
My problem is that I know it can be solved but I can't use the Chinese Remainder Theorem because for example $(3,6)\neq1$... can anyone help?
 A: The equivalence $x\equiv2\pmod4$ implies $x\equiv0\pmod2$.
The equivalence $x\equiv0\pmod2$ and $x\equiv2\pmod3$ imply that $x\equiv2\pmod6$.
Thus, of the $4$ equivalences, we only need to use $x\equiv2\pmod4$ and $x\equiv2\pmod3$.
Therefore, we need to solve the equivalences
$$
x\equiv2\pmod3\\
x\equiv2\pmod4\\
x\equiv2\pmod5
$$
Hint: What do these say about $x-2$?
A: Note that $\mbox{lcm}(2,3,4,5,6) = 60.$  Change your system slightly to 
$$
x \equiv 0\mod2\\ 
x \equiv 0\mod3\\
x \equiv 0\mod4\\
x \equiv 0\mod5\\
x \equiv 0\mod6\\
$$
so that the solution is obviously $x = 60k,$ where $k$ is any integer.  Then observe that adding $2$ to any of these solutions, must be a solution to the original system.  So the set of all solutions is $60k+2.$
A: Since $x \equiv 0 \mod 2$, $x \in \{2, 4, 6, 8, ...\}$
Since $x \equiv 2 \mod 3$, $x \in \{2, 8, 14, ...\}$
Since $x \equiv 2 \mod 4$, $x \in \{2, 14, 26, ...\}$
Since $x \equiv 2 \mod 5$, $x \in \{2, 62, 122, ...\}$
The last condition is already covered by the first two equivalences.
The answer to the question is 122.
A: $x$ satisfies the original set of equations if and only if it satisfies the equations $x\equiv 2\pmod 3$, $x\equiv 2\pmod 4$ and $x\equiv 2\pmod 5$. Now you can apply the Chinese remainder theorem because these moduli are pairwise coprime.
