Difference between revisions of "2024 AMC 8 Problems/Problem 1"
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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): | 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> | <cmath>12-2-(2+2+2+2)=10-8=2</cmath> | ||
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==Video Solution (easy to understand)== | ==Video Solution (easy to understand)== | ||
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~Math-X | ~Math-X | ||
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==Video Solution by Interstigation== | ==Video Solution by Interstigation== | ||
https://youtu.be/ktzijuZtDas&t=36 | https://youtu.be/ktzijuZtDas&t=36 | ||
Revision as of 16:27, 11 February 2024
Contents
Problem
What is the ones digit of![\[222,222-22,222-2,222-222-22-2?\]](http://latex.artofproblemsolving.com/e/2/f/e2fbba6a2cccb3d60b70444a8b272999a204281e.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.
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, 
ends in , 
ends in 
, 
ends in 
, 
ends in 
, and 
ends in 
.
~iasdjfpawregh ~hockey
Solution 4
Let 
be equal to the expression at hand. We reduce each term modulo 
to find the units digit of each term in the expression, and thus the units digit of the entire thing:
![\[S\equiv 2 - 2 - 2 - 2- 2- 2 \equiv -8 \equiv -8 + 10\equiv \boxed{\textbf{(B) } 2} \pmod{10}.\]](http://latex.artofproblemsolving.com/3/9/d/39dd43bbf71a3102bc3e6addcd0fa9bd9a724945.png)
-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):
![\[12-2-(2+2+2+2)=10-8=2\]](http://latex.artofproblemsolving.com/0/4/1/041203878d9ebe56721f48cbed9e09abe44cf179.png)
Thus, we get the answer 
- 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
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. 
