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.