Example 1. Find the last three digits of $769^{304}$.

 untranslated text remains in its original form, preserving the line breaks and format.
Solution: Since $769^{304}=769^{301} \times 769^{3}$, applying Theorem Six we get
$769^{301}=769^{4 \times 8^{2} \times 3+1}$, whose last three digits are 769.
It is not difficult to calculate that the last three digits of $769^{8}$ are 609.
Therefore, the last three digits determined by $769 \times 609$ are 321,
which means the last three digits of $769^{304}$ are 321.