Calculate the digit of the order of tens and hundreds of expansions of:
$7^{100}$ and $11^{2004}$
I don't know a faster method than $7 ^3\equiv 43$ (mod100) and $11 ^ 2 \equiv 21 $(mod 100), and keep reducing by multiplying until you get the result. I think this method would take a while [mod (100) and (mod 1.000)]
Should this really be laborious?
The trick is to find a power which is congruent to $1$.
Note that $7^{20}\equiv 1$ mod $1000$ and $11^{50}\equiv 1$ mod $1000$.
Therefore $7^{100}\equiv 1$ mod $1000$.
Can you now find $11^{2004}$?