levensailorlm: What are the last two digits in 3^2010?
Okay, there is 2 ways of doing this, you culd find a high powered online calculator and just get the answer. I don't suppose that this is what you had in mind?
or
If I find a value x such that 3
x ends in 01 this will help a lot, because any number that is multiplied by another number ending in 01 will keep the same last 2 digits. (Think about it with long multiplication to see that this is true.)
I'm going to write down the first powers of 3 and look at the pattern that they make. Numbers are full of patterns
3, 9, 27, 81,
243, 729, 2187, ....61,
...83, ...49, ...47, ...41
so
3
4 = 81, 3
8=...61, 3
12=...41
Looking at this pattern, I am guessing that 3
16= ...21, 3
20 = ...01
I'll check that on my ordinary calculator. 3
20 = 3,486,784,401 GOOD, Just as I predicted.
So
3
2010 = 3
20*100+10 = 3
20*100 * 3
10 = (3
20)
100 * 3
10 = some number ending in 01 * 59049
= some number ending in 49
Now I will run it through a powerful calculator to see if my answer is correct.
( the web 2.0 calc won't do this so I had to look elsewhere)
+118654 
