Difference between revisions of "2016 AMC 10B Problems/Problem 8"
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In a very similar fashion, we find that <math>2015^{2016} \equiv 15^{2016} \pmod{100}</math>, which equals <math>225^{1008}</math>. Next, since every power (greater than <math>0</math>) of every number ending in <math>25</math> will end in <math>25</math> (which can easily be verified), we get <math>225^{1008} \equiv 25 \pmod{100}</math>. (In this way, we don't have to worry about the exponent very much.) Finally, <math>2017 \equiv 17 \pmod{100}</math>, and thus <math>2015^{2016}-2017 \equiv 25-17 \equiv 08 \pmod{100}</math>, as above. | In a very similar fashion, we find that <math>2015^{2016} \equiv 15^{2016} \pmod{100}</math>, which equals <math>225^{1008}</math>. Next, since every power (greater than <math>0</math>) of every number ending in <math>25</math> will end in <math>25</math> (which can easily be verified), we get <math>225^{1008} \equiv 25 \pmod{100}</math>. (In this way, we don't have to worry about the exponent very much.) Finally, <math>2017 \equiv 17 \pmod{100}</math>, and thus <math>2015^{2016}-2017 \equiv 25-17 \equiv 08 \pmod{100}</math>, as above. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/GVPF6CcCugU | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2016|ab=B|num-b=7|num-a=9}} | {{AMC10 box|year=2016|ab=B|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 11:29, 16 July 2020
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
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 |
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