Question 100
Let $N \gt 1$ be an integer, and let $d_1 \lt \dots \lt d_s$ be all its positive divisors (thus, $d_1 = 1$ and $d_s = N$). Find all possible values of $N$ for which
$$
\gcd(d_1, d_2) + \gcd(d_2, d_3) + \dots + \gcd(d_{s-1}, d_s) = N - 2. \quad (\text{A. Kuznetsov})
$$
$$
\gcd(d_1, d_2) + \gcd(d_2, d_3) + \dots + \gcd(d_{s-1}, d_s) = N - 2. \quad (\text{A. Kuznetsov})
$$