Difference between revisions of "2021 AMC 12A Problems/Problem 3"
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
| − | ==Video Solution (Simple)== | + | ==Video Solutions== |
| + | ===Video Solution (Simple)=== | ||
https://youtu.be/SEp9flDYm2c | https://youtu.be/SEp9flDYm2c | ||
~ Education, the study Of Everything | ~ Education, the study Of Everything | ||
| − | == | + | ===by North America Math Contest Go Go Go=== |
https://www.youtube.com/watch?v=hMqA8i8a2SQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=3 | https://www.youtube.com/watch?v=hMqA8i8a2SQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=3 | ||
| − | == | + | ===by Aaron He=== |
https://www.youtube.com/watch?v=xTGDKBthWsw&t=1m28s | https://www.youtube.com/watch?v=xTGDKBthWsw&t=1m28s | ||
| − | == | + | ===by Punxsutawney Phil=== |
https://youtube.com/watch?v=MUHja8TpKGw&t=143s | https://youtube.com/watch?v=MUHja8TpKGw&t=143s | ||
| − | == | + | ===by Hawk Math=== |
https://www.youtube.com/watch?v=P5al76DxyHY | https://www.youtube.com/watch?v=P5al76DxyHY | ||
| − | == | + | ===(Using Algebra and Meta-solving)=== |
https://youtu.be/d2musztzDjw | https://youtu.be/d2musztzDjw | ||
-pi_is_3.14 | -pi_is_3.14 | ||
| − | == | + | ===by WhyMath=== |
https://youtu.be/VpYmQEKcBpA | https://youtu.be/VpYmQEKcBpA | ||
~savannahsolver | ~savannahsolver | ||
| − | == | + | ===by TheBeautyofMath=== |
https://youtu.be/50CThrk3RcM?t=107 (for AMC 10A) | https://youtu.be/50CThrk3RcM?t=107 (for AMC 10A) | ||
| + | ===by IceMatrix=== | ||
https://youtu.be/rEWS75W0Q54?t=198 (for AMC 12A) | https://youtu.be/rEWS75W0Q54?t=198 (for AMC 12A) | ||
~IceMatrix | ~IceMatrix | ||
| − | == | + | ===(Problems 1-3)=== |
https://youtu.be/CupJpUzKPB0 | https://youtu.be/CupJpUzKPB0 | ||
~MathWithPi | ~MathWithPi | ||
| − | == | + | ===by The Learning Royal=== |
https://youtu.be/slVBYmcDMOI | https://youtu.be/slVBYmcDMOI | ||
Revision as of 07:49, 18 August 2021
- The following problem is from both the 2021 AMC 10A #3 and 2021 AMC 12A #3, so both problems redirect to this page.
Contents
Problem
The sum of two natural numbers is 
. One of the two numbers is divisible by 
. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Solution 1
The units digit of a multiple of will always be 
. We add a whenever we multiply by . So, removing the units digit is equal to dividing by .
Let the smaller number (the one we get after removing the units digit) be 
. This means the bigger number would be 
.
We know the sum is 
so 
. So 
. The difference is 
. So, the answer is 
.
--abhinavg0627
Solution 2 (Lazy Speed)
Since the ones place of a multiple of is , this implies the other integer has to end with a 
since both integers sum up to a number that ends with a . Thus, the ones place of the difference has to be 
, and the only answer choice that ends with an 
is 
~CoolJupiter 2021
Another quick solution is to realize that the sum is represents a number 
added to 
. The difference is 
, which is 
of the given sum.
Solution 3 (Vertical Addition and Logic)
Let the larger number be 
It follows that the smaller number is 
Adding vertically, we have
![\[\begin{array}{cccccc} & A & B & C & D & 0 \\ + & & A & B & C & D \\ \hline & & & & & \\ [-2.5ex] & 1 & 7 & 4 & 0 & 2 \\ \end{array}\]](http://latex.artofproblemsolving.com/5/3/9/539e47c35998235510df757bc1f6fdea5091797b.png)
Working from right to left, we get ![\[D=2\implies C=8 \implies B=5 \implies A=1.\]](http://latex.artofproblemsolving.com/b/6/1/b61dd6a92108f96ce8b9940a52e71cc0d5c86228.png)
The larger number is 
and the smaller number is 
Their difference is 
~MRENTHUSIASM
Video Solutions
Video Solution (Simple)
~ Education, the study Of Everything
by North America Math Contest Go Go Go
https://www.youtube.com/watch?v=hMqA8i8a2SQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=3
by Aaron He
https://www.youtube.com/watch?v=xTGDKBthWsw&t=1m28s
by Punxsutawney Phil
https://youtube.com/watch?v=MUHja8TpKGw&t=143s
by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
(Using Algebra and Meta-solving)
-pi_is_3.14
by WhyMath
~savannahsolver
by TheBeautyofMath
https://youtu.be/50CThrk3RcM?t=107 (for AMC 10A)
by IceMatrix
https://youtu.be/rEWS75W0Q54?t=198 (for AMC 12A)
~IceMatrix
(Problems 1-3)
~MathWithPi
by The Learning Royal
See also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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 AMC 10 Problems and Solutions | ||
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 2 |
Followed by Problem 4 |
| 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 AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
