2021 AMC 12A Problems/Problem 3
- The following problem is from both the 2021 AMC 10A #3 and 2021 AMC 12A #3, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Solution 1 (Algebra)
- 3 Solution 2 (Arithmetic)
- 4 Solution 3 (Vertical Addition and Logic)
- 5 Solution 4 (Logic)
- 6 Video Solutions
- 6.1 Video Solution (Simple)
- 6.2 Video Solution by North America Math Contest Go Go Go
- 6.3 Video Solution by Aaron He
- 6.4 Video Solution by Punxsutawney Phil
- 6.5 Video Solution by Hawk Math
- 6.6 Video Solution by OmegaLearn (Using Algebra and Meta-solving)
- 6.7 Video Solution by WhyMath
- 6.8 Video Solution by TheBeautyofMath
- 6.9 Video Solution by IceMatrix
- 6.10 Video Solution (Problems 1-3)
- 6.11 Video Solution by The Learning Royal
- 7 See also
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 (Algebra)
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 (Arithmetic)
Since the unit's place of a multiple of is , the other integer must end with a , for both integers sum up to a number ending in a . Thus, the unit's place of the difference must be , and the only answer choice that ends with an is .
Another quick solution is to realize that the sum represents a number added to . The difference is , which is of the given sum.
~CoolJupiter 2021
~Extremelysupercooldude (Minor grammar edits)
Solution 3 (Vertical Addition and Logic)
Let the larger number be It follows that the smaller number is Adding vertically, we have Working from right to left, we get The larger number is and the smaller number is Their difference is
~MRENTHUSIASM
Solution 4 (Logic)
We know that the larger number has a units digit of since it is divisible by 10. If is the ten's digit of the larger number, then is the units digit of the smaller number. Since the sum of the natural numbers has a unit's digit of , .
The units digit of the larger number is and the units digit of the smaller number is , so the positive difference between the numbers is 8. There is only one answer choice that has this units digit, and that is
(Similar to MRENTHUSIASM's solution)
~AARUSHGORADIA18
Video Solutions
Video Solution (Simple)
~ Education, the study Of Everything
Video Solution by North America Math Contest Go Go Go
https://www.youtube.com/watch?v=hMqA8i8a2SQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=3
Video Solution by Aaron He
https://www.youtube.com/watch?v=xTGDKBthWsw&t=1m28s
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=MUHja8TpKGw&t=143s
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn (Using Algebra and Meta-solving)
-pi_is_3.14
Video Solution by WhyMath
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/50CThrk3RcM?t=107 (for AMC 10A)
Video Solution by IceMatrix
https://youtu.be/rEWS75W0Q54?t=198 (for AMC 12A)
~IceMatrix
Video Solution (Problems 1-3)
~MathWithPi
Video Solution 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.