Difference between revisions of "2011 AMC 8 Problems/Problem 22"
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<math> \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7 </math> | <math> \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7 </math> | ||
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==Solution 1== | ==Solution 1== | ||
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Since we want the tens digit, we can find the last two digits of <math>7^{2011}</math>. We can do this by using modular arithmetic. | Since we want the tens digit, we can find the last two digits of <math>7^{2011}</math>. We can do this by using modular arithmetic. | ||
− | <cmath>7\equiv 07 \pmod{100}.</cmath> | + | <cmath>7^1\equiv 07 \pmod{100}.</cmath> |
<cmath>7^2\equiv 49 \pmod{100}.</cmath> | <cmath>7^2\equiv 49 \pmod{100}.</cmath> | ||
<cmath>7^3\equiv 43 \pmod{100}.</cmath> | <cmath>7^3\equiv 43 \pmod{100}.</cmath> | ||
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<cmath>7^{2011}\equiv (7^4)^{502}\times 7^3\equiv 7^3\equiv 343\equiv 43\pmod{100}.</cmath> | <cmath>7^{2011}\equiv (7^4)^{502}\times 7^3\equiv 7^3\equiv 343\equiv 43\pmod{100}.</cmath> | ||
From the above, we can conclude that the last two digits of <math>7^{2011}</math> are 43. Since they have asked us to find the tens digit, our answer is <math>\boxed{\textbf{(D)}\ 4}</math>. | From the above, we can conclude that the last two digits of <math>7^{2011}</math> are 43. Since they have asked us to find the tens digit, our answer is <math>\boxed{\textbf{(D)}\ 4}</math>. | ||
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+ | -Ilovefruits | ||
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+ | ==Video Solution by OmegaLearn== | ||
+ | https://youtu.be/7an5wU9Q5hk?t=1710 | ||
==Video Solution== | ==Video Solution== | ||
− | https://youtu.be/ | + | https://youtu.be/lxtYmUzQQ8w ~David |
==See Also== | ==See Also== | ||
{{AMC8 box|year=2011|num-b=21|num-a=23}} | {{AMC8 box|year=2011|num-b=21|num-a=23}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 11:34, 15 July 2024
Problem
What is the tens digit of ?
Solution 1
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
.
-Ilovefruits
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.