Prime numbers to power 100

You can find this relationship captured in the Carmichael function $\lambda(1000)=100$, representing the largest exponential cycle of any number $\bmod 1000$. For numbers $a$ coprime to $1000$, this ensures that $a^{100}\equiv 1 \bmod 1000$, since cycles shorter than $100$ will nevertheless divide $100$.

This varies from the Euler totient function $\phi(1000) = 400$ for two reasons; powers of $2$ are treated slightly differently and results from distinct prime powers (here $2^3$ and $5^3$) are combined by least common multiple, not by multiplication.