2024 AMC 8 Problems/Problem 1

Problem

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.

i am smart

~ Dreamer1297

Solution 2(Tedious)

Using the oniichan Thereom, we deduce that the answer is (B)

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 $\boxed{\textbf{(B) } 2}$ .

~iasdjfpawregh ~hockey

Solution 4

Let $S$ be equal to the expression at hand. We reduce each term modulo $10$ to find the units digit of each term in the expression, and thus the units digit of the entire thing:

$S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.$

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

- FU-King

Video Solution (easy to understand)

https://youtu.be/BaE00H2SHQM?si=_8lhp8-dzNxZ-eUQ

~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.

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