Difference between revisions of "2019 AMC 10A Problems/Problem 2"
(→Solution 4) |
(→Solution 4) |
||
(4 intermediate revisions by 3 users not shown) | |||
Line 17: | Line 17: | ||
~savannahsolver | ~savannahsolver | ||
− | == Video Solution == | + | == Video Solution 3 by OmegaLearn == |
https://youtu.be/zfChnbMGLVQ?t=3899 | https://youtu.be/zfChnbMGLVQ?t=3899 | ||
Line 23: | Line 23: | ||
==Solution 3== | ==Solution 3== | ||
− | Because we know that <math>5^3</math> is a factor of <math>15!</math> and <math>20!</math>, the last three digits of both numbers is a 0, this means that the difference of the hundreds digits is also <math>\boxed{\ | + | Because we know that <math>5^3</math> is a factor of <math>15!</math> and <math>20!</math>, the last three digits of both numbers is a 0, this means that the difference of the hundreds digits is also <math>\boxed{\textbf{(A) }0}</math>. |
− | |||
==Solution 4== | ==Solution 4== | ||
− | We can clearly see that <math>20! \equiv 15! \equiv 0 \pmod{ | + | We can clearly see that <math>20! \equiv 15! \equiv 0 \pmod{1000}</math>, so <math>20! - 15! \equiv 0 \pmod{100}</math> meaning that the last two digits are equal to <math>00</math> and the hundreds digit is <math>0</math>, or <math>\boxed{\textbf{(A)}\ 0}</math>. |
--abhinavg0627 | --abhinavg0627 | ||
Line 37: | Line 36: | ||
<math>15!= 1307674368000</math> | <math>15!= 1307674368000</math> | ||
− | Then, we see that the hundred digit is <math>0-0=0</math> | + | Then, we see that the hundred digit is <math>0-0=\boxed{\textbf{(A)}\ 0}</math>. |
+ | |||
+ | this was the indended solution for this question | ||
-dragoon | -dragoon |
Latest revision as of 12:06, 26 December 2023
Contents
Problem
What is the hundreds digit of
Video Solution 1
Education, The Study of Everything
Video Solution 2
~savannahsolver
Video Solution 3 by OmegaLearn
https://youtu.be/zfChnbMGLVQ?t=3899
~ pi_is_3.14
Solution 3
Because we know that is a factor of and , the last three digits of both numbers is a 0, this means that the difference of the hundreds digits is also .
Solution 4
We can clearly see that , so meaning that the last two digits are equal to and the hundreds digit is , or .
--abhinavg0627
Solution 5 (Brute Force)
Then, we see that the hundred digit is .
this was the indended solution for this question
-dragoon
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.