Difference between revisions of "2019 AMC 10A Problems/Problem 2"
Zygarde6758 (talk | contribs) m (→Solution 3 (Brute Force)) |
|||
| (38 intermediate revisions by 19 users not shown) | |||
| Line 4: | Line 4: | ||
<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 == | + | ==Solution 1== |
| + | 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 <math>0</math>, this means that the difference of the hundreds digits is also <math>\boxed{\textbf{(A) }0}</math>. | ||
| − | + | ==Solution 2== | |
| + | |||
| + | 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>\boxed{\textbf{(A)}\ 0}</math>. | ||
| + | |||
| + | --abhinavg0627 | ||
| + | |||
| + | ==Solution 3 (Brute Force)== | ||
| + | |||
| + | <math>20!= 2432902008176640000</math> | ||
| + | <math>15!= 1307674368000</math> | ||
| + | |||
| + | Then, we see that the hundreds digit is <math>0-0=\boxed{\textbf{(A)}\ 0}</math>. | ||
| + | |||
| + | ~dragoon | ||
| + | |||
| + | Please do not do this and only use this solution as a last resort. | ||
| + | |||
| + | ==Video Solution by Education, the Study of Everything== | ||
| + | |||
| + | https://youtu.be/J4Bqztwjyxw | ||
| + | |||
| + | ~Education, The Study of Everything | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | |||
| + | https://youtu.be/V1fY0oLSHvo | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution by OmegaLearn== | ||
| + | |||
| + | https://youtu.be/zfChnbMGLVQ?t=3899 | ||
| + | |||
| + | ~pi_is_3.14 | ||
==See Also== | ==See Also== | ||
Latest revision as of 18:14, 4 November 2024
Contents
Problem
What is the hundreds digit of 
Solution 1
Because we know that 
is a factor of 
and 
, the last three digits of both numbers is a 
, this means that the difference of the hundreds digits is also 
.
Solution 2
We can clearly see that 
, so 
meaning that the last two digits are equal to 
and the hundreds digit is 
.
--abhinavg0627
Solution 3 (Brute Force)
Then, we see that the hundreds digit is 
.
~dragoon
Please do not do this and only use this solution as a last resort.
Video Solution by Education, the Study of Everything
~Education, The Study of Everything
Video Solution by WhyMath
~savannahsolver
Video Solution by OmegaLearn
https://youtu.be/zfChnbMGLVQ?t=3899
~pi_is_3.14
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. 
