2000 AMC 8 Problems/Problem 14
Contents
Problem
What is the units digit of ?
Video Solution by OmegaLearn
https://youtu.be/7an5wU9Q5hk?t=1552
Solution
Finding a pattern for each half of the sum, even powers of have a units digit of , and odd powers of have a units digit of . So, has a units digit of .
Powers of have the exact same property, so also has a units digit of . which has a units digit of , so the answer is .
Solution 2
Using modular arithmetic:
Similarly,
We have
-ryjs
Solution 3
Experimentation gives
Using this we have
Both and are odd, so we are left with which has units digit -ryjs
Solution 4
36972963764972677265718790562880544059566876428174110243025997242355257045527752342141065001012823272794097888 9548326540119429996769494359451621570193644014418071060667659303363419435659472789623878 which ends in 8. :-) If you are really stupid and have time (like about 3 hours), this is a real brainless bash.
--hefei417
See Also
2000 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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