Solution 0d8469c0-87f9-4ae0-9b98-1574a03f6ba9

Points: 

Let
\[
S=\{2,3,4,5,6\}.
\]
Then
\[
\operatorname{lcm} S=60.
\]

Lemma 1: $N-1=60k$ for some nonnegative integer $k$

For all $i\in S$, we have
\[
N\equiv 1 \pmod{i},
\]
meaning
\[
i\mid N-1.
\]
So
\[
N-1=k\cdot \operatorname{lcm} S=60k.
\]

Lemma 2: The minimum value $N=1$ occurs at $k=0$

We know
\[
N=60k+1>0,
\]
and we see that the possible values $P$ are
\[
P=\{1,61,121,\ldots\}.
\]
But since we must minimize $N$, we find
\[
N=\min P=1.
\]

Final answer:
N=1