<s>[INST] Intermediate Modular Arithmetic Strategy And Questions I have these problems and I know how to solve ones with two equations but not more.  Can anyone provide solutions?  Thanks in advance!

Find the smallest positive integer that satisfies the system of congruences
\begin{align*}
N &\equiv 1 \pmod{7}, \\
N &\equiv 7 \pmod{13}, \\
N &\equiv 13 \pmod{20}.
\end{align*}

Find the smallest positive $N$ such that
\begin{align*}
N &\equiv 6 \pmod{12}, \\
N &\equiv 6 \pmod{18}, \\
N &\equiv 6 \pmod{24}, \\
N &\equiv 6 \pmod{30}, \\
N &\equiv 6 \pmod{60}.
\end{align*}

How many positive integers less than or equal to $6\cdot7\cdot8\cdot9$ solve the system of congruences
\begin{align*}
m &\equiv 5 \pmod{6}, \\
m &\equiv 4 \pmod{7}, \\
m &\equiv 3 \pmod{8}, \\
m &\equiv 3 \pmod{9}.
\end{align*}
 [/INST] For problem 1:
$$N=7p+1$$
$$7p+1 \equiv 7 \pmod{13}$$
$$7p \equiv 6 \pmod{13}$$
$$p \equiv 12 \pmod{13}$$
(This is calculated by finding the inverse of 7 mod 13).
$$N \equiv 7*12+1 \pmod{13*7}$$
$$N \equiv 85 \pmod{91}$$
$$N = 91q + 85$$
$$N \equiv 13 \pmod{20} $$
$$91q + 85 \equiv 13 \pmod{20} $$
$$91q \equiv 8 \pmod{20} $$
$$11q \equiv 8 \pmod{20}$$
$$q \equiv 8 \pmod{20} $$
$$N \equiv 91*8+85 \pmod{91*20}$$
$$N \equiv 813 mod \pmod{1820}$$
$$N = 813$$
For problem 2, $N = 6$ is the obvious solution.
For problem 3, we see that it is not possible to have $N \equiv 5 \pmod{6}$ and $N \equiv 3 \pmod{9}$ (first suggests $N$ is not divisible by 3, second suggests $N$ is divisible by 3). So, here there are 0 valid solutions.
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