00:01
So in this problem, we have to find the last three digits of 9 to the 105 using the chinese remainder theorem.
00:22
So really, what does this mean? this means that we take 9 to the 105th power is going to be x times mod or x mod 1 ,000.
00:40
Okay.
00:41
Well, we know that 9 cubed is equivalent to 729 mod 1 ,000.
01:02
Why do we know that? because 9 squared is 81, and so 9 cubed is 9, right, is 729.
01:14
Because 81 times 9 would be 729.
01:17
And that's less than 1 ,000, so it's the same number.
01:21
Then we know 9 of the 5th, well, that'll be 729 times 81, wouldn't it? this is going to end up being 49 modulus 1 ,000.
01:43
Okay...