Difference between revisions of "2019 AMC 10A Problems/Problem 2"
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<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5</math> | ||
| − | == Solution == | + | ==Video Solution 1== |
| − | + | https://youtu.be/J4Bqztwjyxw | |
| − | + | Education, The Study of Everything | |
| − | |||
| − | ==Video Solution== | + | |
| + | ==Video Solution 2== | ||
https://youtu.be/V1fY0oLSHvo | https://youtu.be/V1fY0oLSHvo | ||
~savannahsolver | ~savannahsolver | ||
| + | |||
| + | == Video Solution 3 by OmegaLearn == | ||
| + | https://youtu.be/zfChnbMGLVQ?t=3899 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| + | ==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{\textbf{(A) }0}</math>. | ||
| + | |||
| + | ==Solution 4== | ||
| + | |||
| + | 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 | ||
| + | |||
| + | ==Solution 5 (Brute Force)== | ||
| + | |||
| + | <math>20!= 2432902008176640000</math> | ||
| + | <math>15!= 1307674368000</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 | ||
| + | |||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2019|ab=A|num-b=1|num-a=3}} | {{AMC10 box|year=2019|ab=A|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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. 
