2012 AMC 8 Problems/Problem 12
Contents
Problem
What is the units digit of ?
Videos
https://youtu.be/7an5wU9Q5hk?t=1186
https://youtu.be/6c_s967T7cA ~savannahsolver
Solution 1
The problem wants us to find the units digit of , therefore, we can eliminate the tens digit of
, because the tens digit will not affect the final result. So our new expression is
. Now we need to look for a pattern in the units digit.
We observe that there is a pattern for the units digit which recurs every four powers of three. Using this pattern, we can subtract 1 from 2012 and divide by 4. The remainder is the power of three that we are looking for, plus one. divided by
leaves a remainder of
, so the answer is the units digit of
, or
. Thus, we find that the units digit of
is
.
Solution 2
Ignore the tens digit of , we find a pattern in the units digit that
. We also find
can be divided by
evenly, which is
. So
=
. Because the units digit of
,so the units digit
. Thus, the units digit of
is
. ---LarryFlora
See Also
2012 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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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