Difference between revisions of "2024 AMC 10A Problems/Problem 1"
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\end{align*}</cmath> | \end{align*}</cmath> | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
| + | == Solution 3 (Quickest Way) == | ||
| + | |||
| + | We simply look at the units digit of the problem we have(or take mod 10) | ||
| + | <cmath>9901\cdot101-99\cdot10101 \equiv 1*1 - 9*1 = 2 \mod{10}.</cmath> | ||
| + | Since the only answer with 2 in the units digit is <math>\boxed{\textbf{(A)}}</math> we are done! | ||
==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|before=First Problem|num-a=2}} | {{AMC10 box|year=2024|ab=A|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 15:27, 8 November 2024
Contents
Problem
What is the value of ![\[9901\cdot101-99\cdot10101?\]](http://latex.artofproblemsolving.com/1/3/4/134a62dd445139bc0163af4dc4a59a4de7086e6e.png)
Solution 1 (Direct Computation)
The likely fastest method will be direct computation. 
evaluates to 
and 
evaluates to 
. The difference is 
Solution by juwushu.
Solution 2 (Distributive Property)
We have
~MRENTHUSIASM
Solution 3 (Quickest Way)
We simply look at the units digit of the problem we have(or take mod 10)
![\[9901\cdot101-99\cdot10101 \equiv 1*1 - 9*1 = 2 \mod{10}.\]](http://latex.artofproblemsolving.com/c/4/e/c4e379c4a9c5245b85a23381d38befa04f8eaab5.png)
Since the only answer with 2 in the units digit is 
we are done!
See also
| 2024 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 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. 
