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

Latest revision as of 07:58, 25 November 2024

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: $222{,}222-22{,}222-2{,}222-222-22-2?$ $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

Solution 1

We can rewrite the expression as $222,222-(22,222+2,222+222+22+2).$

We note that the units digit of the addition is $0$ because all the units digits of the five numbers are $2$ and $5*2=10$ , which has a units digit of $0$ .

Now, we have something with a units digit of $0$ subtracted from $222,222$ . The units digit of this expression is obviously $2$ , and we get $\boxed{B}$ as our answer.

Solution 2

$222,222-22,222 = 200,000$ $200,000 - 2,222 = 197778$ $197778 - 222 = 197556$ $197556 - 22 = 197534$ $197534 - 2 = 1957532$ So our answer is $\boxed{\textbf{(B) } 2}$ .

Solution 3

We only care about the unit's 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 $\boxed{\textbf{(B) } 2}$ .

-unknown

minor edits by EzLx

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): $(12-2)-(2+2+2+2)=10-8=2$ Thus, we get the answer $\boxed{(B)}$

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!)

https://youtu.be/Ol1seWX0xHY

~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

https://youtu.be/RzPadkHd3Yc

Video Solution by WhyMath

https://youtu.be/i4mcj3jRTxM

@@ Line 1: / Line 1: @@
-==Problem==
+==Problem 1==
-What is the ones digit of <cmath>222,222-22,222-2,222-222-22-2?</cmath>
+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>
+==Solution 1==
+We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</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>.
+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.
 ==Solution 2==
-We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2)</cmath>.
+<math>222,222-22,222 = 200,000</math>
+<math>200,000 - 2,222 = 197778</math>
+<math>197778 - 222 = 197556</math>
+<math>197556 - 22 = 197534</math>
+<math>197534 - 2 = 1957532
+</math>
+So our answer is <math>\boxed{\textbf{(B) } 2}</math>.
+==Solution 3==
+We only care about the unit's 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>.
+-unknown
+minor edits by EzLx
+==Solution 4==
-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 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>
+==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!)==
+https://youtu.be/Ol1seWX0xHY
+~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
-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.
+==Video Solution by Dr. David==
-~ Dreamer1297
+https://youtu.be/RzPadkHd3Yc
-==Solution 1==
+==Video Solution by WhyMath==
+https://youtu.be/i4mcj3jRTxM
-<cmath>222,222-22,222-2,222-222-22-2</cmath>
+==See Also==
-<cmath> = 200,000 - 2,222 - 222 - 22 - 2 </cmath>
+{{AMC8 box|year=2024|before=First Problem|num-a=2}}
-<cmath> = 197778 - 222 - 22 - 2 </cmath>
+{{MAA Notice}}
-<cmath> = 197556 - 22 - 2 </cmath>
-<cmath> = 197534 - 2 </cmath>
-<cmath> = 197532 </cmath>
-This means the ones digit is <math>\boxed{(B) \hspace{1 mm} 2}</math>
-<math>\newline</math>
-~ nikhil
-~ CXP

2024 AMC 8 (Problems • Answer Key • Resources)
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