Difference between revisions of "2024 AMC 8 Problems/Problem 1"

Latest revision as of 08:46, 16 February 2025

1 Problem
2 Solution 1
3 Solution 2
4 Solution 3
5 Solution 4
6 Solution 5
7 Solution 6
8 Video Solution 1 (Detailed Explanation) 🚀⚡📊
9 Video by MathTalks_Now
10 Video Solution by Central Valley Math Circle (Goes through full thought process)
11 Video Solution 2 (MATH-X)
12 Video Solution 3 (A Clever Explanation You’ll Get Instantly)
13 Video Solution 4 (Quick and Easy)
14 Video Solution 5 Interstigation
15 Video Solution 6 Daily Dose of Math
16 Video Solution 7 Dr. David
17 Video Solution 8 WhyMath
18 See Also

Problem

What is the last digit of: $222{,}222-22{,}222-2{,}222-222-22-2?$ $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10$

Solution 1

We can rewrite the expression as $222,222-(22,222+2,222+222+22+2)$ . We note that the units digit of $22,222+2,222+222+22+2$ is $0$ because all the units digits of the five numbers are $2$ and $5\cdot2=10$ , which has a units digit of $0$ . Now, we have something with a units digit of $0$ subtracted from $222,222$ , and so the units digit of this expression is $\boxed{\textbf{(B) } 2}$ .

Solution 2

$222,222-22,222 = 200,000$ $200,000 - 2,222 = 197778$ $197778 - 222 = 197556$ $197556 - 22 = 197534$ $197534 - 2 = 197532$ So our answer is $\boxed{\textbf{(B) } 2}$ .

Solution 3

We only care about the units digits. Thus, $2-2$ ends in $0$ , $0-2$ after regrouping(10-2) ends in $8$ , $8-2$ ends in $6$ , $6-2$ ends in $4$ , and $4-2$ ends in $\boxed{\textbf{(B) } 2}$ .

Solution 4

We just take the units digit of each and subtract, adding an extra ten to the first number so we don't get a negative number: $(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}$

Solution 5

$222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}$

Solution 6

We can ignore the other digits and just do $22-2-2-2-2-2$ . Because you are subtracting five $2s$ and $2\cdot5 = 10$ , you subtract $10$ from $22$ . This gives us 12, so the last digit is $\boxed{\textbf{(B) } 2}$ .

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/jqsbMWhTYRg

~ ChillGuyDoesMath :)

Video by MathTalks_Now

https://www.youtube.com/watch?v=crn37TRMLv4

-rc1219

Video Solution by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/-XcShDyuZIo

Video Solution 2 (MATH-X)

https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130

Video Solution 3 (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53

Video Solution 4 (Quick and Easy)

https://youtu.be/Ol1seWX0xHY

Video Solution 5 Interstigation

https://youtu.be/ktzijuZtDas&t=36

Video Solution 6 Daily Dose of Math

https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR

Video Solution 7 Dr. David

https://youtu.be/RzPadkHd3Yc

Video Solution 8 WhyMath

https://youtu.be/i4mcj3jRTxM

@@ Line 1: / Line 1: @@
 ==Problem==
-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>
+What is the last 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) } 8\qquad\textbf{(E) } 10</math>
 ==Solution 1==
+We can rewrite the expression as <math>222,222-(22,222+2,222+222+22+2)</math>. We note that the units digit of <math>22,222+2,222+222+22+2</math> is <math>0</math> because all the units digits of the five numbers are <math>2</math> and <math>5\cdot2=10</math>, which has a units digit of <math>0</math>. Now, we have something with a units digit of <math>0</math> subtracted from <math>222,222</math>, and so the units digit of this expression is <math>\boxed{\textbf{(B) } 2}</math>.
-We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</cmath>
+==Solution 2==
+<cmath>222,222-22,222 = 200,000</cmath>
-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>.
+<cmath>200,000 - 2,222 = 197778</cmath>
+<cmath>197778 - 222 = 197556</cmath>
-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.
+<cmath>197556 - 22 = 197534</cmath>
+<cmath>197534 - 2 = 197532</cmath>
-i am smart
+So our answer is <math>\boxed{\textbf{(B) } 2}</math>.
-~ Dreamer1297
-==Solution 2(Tedious)==
-Using the oniichan Thereom, we deduce that the answer is (B)
 ==Solution 3==
+We only care about the units digits. 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>.
-We only care about the unit's digits.
-Thus, <math>2-2</math> ends in <math>0</math>, <math>0-2</math> 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>.
-~iasdjfpawregh
-~vockey
 ==Solution 4==
+We just take the units digit of each and subtract, adding an extra ten to the first number so we don't get a negative number:
+<cmath>(12-2)-(2+2+2+2)=10-8=\boxed{\textbf{(B) } 2}</cmath>
-Let <math>S</math> be equal to the expression at hand. We reduce each term modulo <math>10</math> to find the units digit of each term in the expression, and thus the units digit of the entire thing:
+== Solution 5 ==
+<cmath>222{,}222-22{,}222-2{,}222-222-22-2\equiv2-2-2-2-2\equiv-8\equiv\boxed{\textbf{(B) } 2}\pmod{10}</cmath>
-<cmath>S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.</cmath>
--Benedict T (countmath1)
+== Solution 6==
+We can ignore the other digits and just do <math>22-2-2-2-2-2</math>. Because you are subtracting five <math>2s</math> and <math>2\cdot5 = 10</math>, you subtract <math>10</math> from <math>22</math>. This gives us 12, so the last digit is <math>\boxed{\textbf{(B) } 2}</math>.
+== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
+Youtube Link ⬇️
-==Solution 5==
+https://youtu.be/jqsbMWhTYRg
-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):
-<cmath>12-2-(2+2+2+2)=10-8=2</cmath>
-Thus, we get the answer <math>\boxed{(B)}</math>
-- FU-King
+~ ChillGuyDoesMath :)
-==Solution 6(fast)==
+== Video by MathTalks_Now ==
-uwu  <math>\boxed{(uwu)}</math>
-- uwu gamer girl(ꈍᴗꈍ)
+https://www.youtube.com/watch?v=crn37TRMLv4
-This is not useful. Please come up with a proper solution or delete.
+-rc1219
-Hello, please remove this nonsense post, or your account will be in risk of banning.
+==Video Solution by Central Valley Math Circle (Goes through full thought process)==
+https://youtu.be/-XcShDyuZIo
-I DONT CARE NOBODY ASKED(ꈍᴗꈍ)
+==Video Solution 2 (MATH-X)==
+https://youtu.be/BaE00H2SHQM?si=O0O0g7qq9AbhQN9I&t=130
-==Solution 7==
+==Video Solution 3 (A Clever Explanation You’ll Get Instantly)==
--2=0. Therefore, ones digit is the 10th avacado  <math>\boxed{(F)}</math>
+https://youtu.be/5ZIFnqymdDQ?si=IbHepN2ytt7N23pl&t=53
-- iamcalifornia'sresidentidiot
+==Video Solution  4 (Quick and Easy)==
+https://youtu.be/Ol1seWX0xHY
-Hello, please remove this nonsense post, or your account will be in risk of banning
+==Video Solution 5 Interstigation==
+https://youtu.be/ktzijuZtDas&t=36
-this is not nonsense like what are you yapping about this is the most beautiful solution every to be conceived in all of humanity i literally deserve a nobel prize
-PS touch grass
-==Video Solution 1 (easy to digest) by Power Solve==
-https://www.youtube.com/watch?v=dQw4w9WgXcQ
-==Video Solution (easy to understand)==
-https://youtu.be/BaE00H2SHQM?si=_8lhp8-dzNxZ-eUQ
-~Math-X
-==Video Solution by NiuniuMaths (Easy to understand!)==
-https://www.youtube.com/watch?v=dQw4w9WgXcQ
-~Rick Aopsly
+==Video Solution 6 Daily Dose of Math==
+https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
-==Video Solution 2 by uwu==
+==Video Solution 7 Dr. David==
-https://www.youtube.com/watch?v=dQw4w9WgXcQ
+https://youtu.be/RzPadkHd3Yc
-== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
+==Video Solution 8 WhyMath==
+https://youtu.be/i4mcj3jRTxM
-https://www.youtube.com/watch?v=dQw4w9WgXcQ
-== cool solution must see ==
-https://www.youtube.com/watch?v=dQw4w9WgXcQ
-==Video Solution by Interstigation==
-https://youtu.be/ktzijuZtDas&t=36
 ==See Also==
 {{AMC8 box|year=2024|before=First Problem|num-a=2}}
 {{MAA Notice}}

2024 AMC 8 (Problems • Answer Key • Resources)
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