Difference between revisions of "2011 AMC 8 Problems/Problem 22"
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==Solution 1== | ==Solution 1== | ||
We want the tens digit | We want the tens digit | ||
− | So, we take <math>7^{2011}\ (\text{mod }100)</math>. That is congruent to <math>7^{11}\ (\text{mod }100)</math>. From here, it is an easy | + | So, we take <math>7^{2011}\ (\text{mod }100)</math>. That is congruent to <math>7^{11}\ (\text{mod }100)</math>. From here, it is an easy stuff, 7, 49, 43, 01, 07, 49, 43, 01, 07, 49, 43. So the answer is <math>\boxed{\textbf{(D)}\ 4}</math> |
==Solution 2== | ==Solution 2== |
Revision as of 11:51, 21 May 2023
Contents
Problem
What is the tens digit of ?
Solution 1
We want the tens digit
So, we take . That is congruent to
. From here, it is an easy stuff, 7, 49, 43, 01, 07, 49, 43, 01, 07, 49, 43. So the answer is
Solution 2
Since we want the tens digit, we can find the last two digits of . We can do this by using modular arithmetic.
We can write
as
. Using this, we can say:
From the above, we can conclude that the last two digits of
are 43. Since they have asked us to find the tens digit, our answer is
.
~marfu2007
Solution 3
We can use patterns to figure out the answer.
7 to the power of 1 has no tens digit, so we can ignore it.
7 to the power of 2 is 49.
7 to the power of 3 is 343.
7 to the power of 4 is 2401.
7 to the power of 5 is 16807.
By now, we can notice the pattern. The tens digit for 7 to the power of 1 is 0, then 4, then 4, then 0, then 0. It keeps rotating, 2 fours, and 2 zeros in 4 numbers. If we round up for 2011/4, we get 503. 503 * 4 is 2012. So 7 to the 2012 power has a tens digit of 0, since 2012 is a mutiple of 4, and 7 to the power of 4 has a tens digit of 0. We have to subtract a power from 7 to the 2012 power, so the tens digit goes back from 0 to 4 because if we subtract a power from 7 to the power of 4, we have 7 to the power of 3, which has a tens digit of 4. Hence the answer is .
~aarushgoradia18
Video Solution by OmegaLearn
https://youtu.be/7an5wU9Q5hk?t=1710
Video Solution
https://youtu.be/lxtYmUzQQ8w ~David
See Also
2011 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.