Difference between revisions of "2024 AMC 8 Problems/Problem 1"
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==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$ | ==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$ | ||
| + | ==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8<math> | ||
==Solution 1== | ==Solution 1== | ||
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| + | ==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math> | ||
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We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</cmath> | We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</cmath> | ||
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We note that the units digit of the addition is <math>0</math> because all the units digits of the five numbers are <math>2</math> and <math>5*2=10</math>, which has a units digit of <math>0</math>. | We note that the units digit of the addition is <math>0</math> because all the units digits of the five numbers are <math>2</math> and <math>5*2=10</math>, which has a units digit of <math>0</math>. | ||
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Now, we have something with a units digit of <math>0</math> subtracted from <math>222,222</math>. The units digit of this expression is obviously <math>2</math>, and we get <math>\boxed{B}</math> as our answer. | Now, we have something with a units digit of <math>0</math> subtracted from <math>222,222</math>. The units digit of this expression is obviously <math>2</math>, and we get <math>\boxed{B}</math> as our answer. | ||
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i am smart | i am smart | ||
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~ Dreamer12 | ~ Dreamer12 | ||
Revision as of 15:28, 20 March 2024
==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$
==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8
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We can rewrite the expression as ![\[222,222-(22,222+2,222+222+22+2).\]](http://latex.artofproblemsolving.com/d/7/4/d74bfc53ce46f0b91792bb0b074e29716ed43be7.png)
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We note that the units digit of the addition is 
because all the units digits of the five numbers are 
and 
, which has a units digit of .
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Now, we have something with a units digit of subtracted from 
. The units digit of this expression is obviously , and we get 
as our answer.
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i am smart
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~ Dreamer12
Contents
Solution 2 (boring)
222,222-22,222 = 200,000 200,000 - 2,222 = 197778 197778 - 222 = 197556 197556 - 22 = 197534 197534 - 2 = 1957532
So our answer is 
.
Solution 3
We only care about the unit's digits.
Thus, 
ends in , 
ends in 
, 
ends in 
, 
ends in 
, and 
ends in .
~iasdjfpawregh ~hockey
Solution 4
Let 
be equal to the expression at hand. We reduce each term modulo 
to find the units digit of each term in the expression, and thus the units digit of the entire thing:
![\[S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.\]](http://latex.artofproblemsolving.com/3/9/d/39dd43bbf71a3102bc3e6addcd0fa9bd9a724945.png)
-Benedict T (countmath1)
Solution 5
We just take the units digit of each and subtract, or you can do it this way by adding an extra ten to the first number (so we don't get a negative number):
![\[12-2-(2+2+2+2)=10-8=2\]](http://latex.artofproblemsolving.com/0/4/1/041203878d9ebe56721f48cbed9e09abe44cf179.png)
Thus, we get the answer 
- FU-King nice
Video Solution 1 (Quick and Easy!)
~Education, the Study of Everything
Video Solution (easy to understand)
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
~Math-X
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=36
See Also
| 2024 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
