Solution 620ec34c-e7e1-4ab9-a070-ef8ee5aa51a3

Points: 2 pts

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
\[
N\equiv i-1\equiv -1 \pmod{i},
\]
meaning
\[
i\mid N+1.
\]
So
\[
N+1=k\cdot \operatorname{lcm} S=60k.
\]

Lemma 2: The minimum positive value $N=59$ occurs at $k=1$

We know
\[
N=60k-1>0.
\]
The value $k=0$ gives $N=-1$, so it is not possible. Thus the possible positive values $P$ are
\[
P=\{59,119,179,\ldots\}.
\]
Since we must minimize $N$, we find
\[
N=\min P=59.
\]

Final answer:
N=59