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

Latest revision as of 12:02, 20 April 2024

Problem 1

What is the ones digit of $222,22 2-22,222-2,222-222-22-2?$ $\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

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 $0$ , $0-2$ 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, 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 1 (Quick and Easy!)

https://youtu.be/Ol1seWX0xHY

~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

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.

@@ Line 1: / Line 1: @@
-==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$
+==Problem 1==
+What is the ones digit of <cmath>222,22
+-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>
-==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8<math>
 ==Solution 1==
-+
-==Problem==uad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
-−
 −
 We can rewrite the expression as <cmath>222,222-(22,222+2,222+222+22+2).</cmath>
@@ Line 26: / Line 23: @@
 −
-i am smart
-−
-−
-~ Dreamer12
-==Solution 2 (boring)==
+==Solution 2==
 ,222-22,222 = 200,000
@@ Line 50: / Line 40: @@
 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
-~hockey
 ==Solution 4==
-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:
-<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 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):
 <cmath>12-2-(2+2+2+2)=10-8=2</cmath>
 Thus, we get the answer <math>\boxed{(B)}</math>
-- FU-King
-nice
 ==Video Solution 1 (Quick and Easy!)==

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