Difference between revisions of "2016 AMC 10B Problems/Problem 8"
Pi is 3.14 (talk | contribs) (→Video Solution) |
Jackshi2006 (talk | contribs) (→Solution 1) |
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Notice that, for <math>n\ge 2</math>, <math>2015^n\equiv 15^n</math> is congruent to <math>25\pmod{100}</math> when <math>n</math> is even and <math>75\pmod{100}</math> when <math>n</math> is odd. (Check for yourself). Since <math>2016</math> is even, <math>2015^{2016} \equiv 25\pmod{100}</math> and <math>2015^{2016}-2017 \equiv 25 - 17 \equiv \underline{0}8\pmod{100}</math>. | Notice that, for <math>n\ge 2</math>, <math>2015^n\equiv 15^n</math> is congruent to <math>25\pmod{100}</math> when <math>n</math> is even and <math>75\pmod{100}</math> when <math>n</math> is odd. (Check for yourself). Since <math>2016</math> is even, <math>2015^{2016} \equiv 25\pmod{100}</math> and <math>2015^{2016}-2017 \equiv 25 - 17 \equiv \underline{0}8\pmod{100}</math>. | ||
− | So the answer is <math>\textbf{(A)}\ 0</math>. | + | So the answer is <math>\textbf{(A)}\ \boxed{0}</math>. |
− | |||
==Solution 2== | ==Solution 2== |
Revision as of 21:35, 25 January 2021
Problem
What is the tens digit of
Solution 1
Notice that, for ,
is congruent to
when
is even and
when
is odd. (Check for yourself). Since
is even,
and
.
So the answer is .
Solution 2
In a very similar fashion, we find that , which equals
. Next, since every power (greater than
) of every number ending in
will end in
(which can easily be verified), we get
. (In this way, we don't have to worry about the exponent very much.) Finally,
, and thus
, as above.
Video Solution
~savannahsolver
Video Solution
https://youtu.be/zfChnbMGLVQ?t=2683
~ pi_is_3.14
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.