2008 AMC 12A Problems/Problem 15
- The following problem is from both the 2008 AMC 12A #15 and 2008 AMC 10A #24, so both problems redirect to this page.
Problem
Let . What is the units digit of ?
Solution
.
So, . Since is a multiple of four and the units digit of powers of two repeat in cycles of four, .
Therefore, . So the units digit is .
Note
Another way to get is to find the cycles of the digits.
For , we only have to care about the last digit since itself is a two digit and we want the last digit. The last digit of is obviously
For , note that the last digit cycles thru the pattern . (You can easily do this by simply calculating the first powers of .)
Since is a multiple of , the last digit of is evidently
Continue as follows.
~mathboy282
Solution 2 (Video solution)
Video: https://youtu.be/Ib-onAecb1I
See Also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 AMC 12 Problems and Solutions |
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.