TITLE: How to find the last two digits of $299^{33}$. Is there any trick?
QUESTION [10 upvotes]: Find the last two digits of the number $N=299^{33}$

Is there any trick of finding the last two digits of such a big number. If yes, then kindly share it with me
This is a GMAT exam question.

REPLY [3 votes]: You can use the Chinese remainder theorem for these kind of problems.
Solution:
$100 = 4 \times 25$ and $\gcd(4,25) = 1$
Now we know that
$299 \equiv -1 \pmod4$
So, $299^{33}\equiv -1 \pmod4 \equiv 3\pmod4$
Now we also know that
$299 \equiv -1 \pmod{25}$
$299^{33} \equiv -1 \pmod{25}$
Now: $299^{33} \equiv -1 \pmod4$
$299^{33} \equiv -1 \pmod{25}$
so, $299^{33} \equiv -1 \pmod{100} \equiv 99 \pmod{100}$
So the answer is 99