Difference between revisions of "2024 AMC 8 Problems/Problem 1"
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==Problem 1== | ==Problem 1== | ||
| − | What is the ones digit of | + | What is the ones digit of: <cmath>222{,}222-22{,}222-2{,}222-222-22-2?</cmath> |
<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math> | <math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math> | ||
| Line 24: | Line 24: | ||
We only care about the unit's digits. | We only care about the unit's digits. | ||
| − | Thus, <math>2-2</math> ends in <math>0</math>, <math> | + | Thus, <math>2-2</math> ends in <math>0</math>, <math>0-2</math> after regrouping(10-2) ends in <math>8</math>, <math>8-2</math> ends in <math>6</math>, <math>6-2</math> ends in <math>4</math>, and <math>4-2</math> ends in <math>\boxed{\textbf{(B) } 2}</math>. |
-unknown | -unknown | ||
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<cmath>(12-2)-(2+2+2+2)=10-8=2</cmath> | <cmath>(12-2)-(2+2+2+2)=10-8=2</cmath> | ||
Thus, we get the answer <math>\boxed{(B)}</math> | Thus, we get the answer <math>\boxed{(B)}</math> | ||
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| + | ==Video Solution (MATH-X)== | ||
| + | https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130 | ||
| + | |||
| + | ~Math-X | ||
==Video Solution (A Clever Explanation You’ll Get Instantly)== | ==Video Solution (A Clever Explanation You’ll Get Instantly)== | ||
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~hsnacademy | ~hsnacademy | ||
| − | ==Video Solution | + | ==Video Solution (Quick and Easy!)== |
https://youtu.be/Ol1seWX0xHY | https://youtu.be/Ol1seWX0xHY | ||
~Education, the Study of Everything | ~Education, the Study of Everything | ||
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==Video Solution by Interstigation== | ==Video Solution by Interstigation== | ||
| Line 63: | Line 63: | ||
https://youtu.be/RzPadkHd3Yc | https://youtu.be/RzPadkHd3Yc | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/i4mcj3jRTxM | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2024|before=First Problem|num-a=2}} | {{AMC8 box|year=2024|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 06:13, 15 November 2024
Contents
- 1 Problem 1
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Video Solution (MATH-X)
- 7 Video Solution (A Clever Explanation You’ll Get Instantly)
- 8 Video Solution (Quick and Easy!)
- 9 Video Solution by Interstigation
- 10 Video Solution by Daily Dose of Math
- 11 Video Solution by Dr. David
- 12 Video Solution by WhyMath
- 13 See Also
Problem 1
What is the ones digit of: ![\[222{,}222-22{,}222-2{,}222-222-22-2?\]](http://latex.artofproblemsolving.com/1/5/6/156c3662fe4c874a80e54887f76c302c13b9d14c.png)
Solution 1
We can rewrite the expression as ![\[222,222-(22,222+2,222+222+22+2).\]](http://latex.artofproblemsolving.com/d/7/4/d74bfc53ce46f0b91792bb0b074e29716ed43be7.png)
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 .
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.
Solution 2
So our answer is 
.
Solution 3
We only care about the unit's digits.
Thus, 
ends in , 
after regrouping(10-2) ends in 
, 
ends in 
, 
ends in 
, and 
ends in .
-unknown
minor edits by Fireball9746
Solution 4
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/1/a/e/1ae83497079c7f1f607247a754db7a749c0380c3.png)
Thus, we get the answer 
Video Solution (MATH-X)
https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
~Math-X
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
~hsnacademy
Video Solution (Quick and Easy!)
~Education, the Study of Everything
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=36
Video Solution by Daily Dose of Math
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
~Thesmartgreekmathdude
Video Solution by Dr. David
Video Solution by WhyMath
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. 
