Product of Totatives-III Divisors of Googol

For a positive integer $n$ , let $\mathbb P(n)$ be the product of all possible positive integers $a \leq n$ with $\gcd(a,n)=1$ .

Find the number of all possible distinct positive divisors $d$ of $10^{100}$ such that $\mathbb P(d) \equiv 1 \pmod{d}$

Suggested Warm-Ups : This and this .