2021 AMC 12A Problems/Problem 3

Problem

The sum of two natural numbers is $17{,}402$ . One of the two numbers is divisible by $10$ . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\qquad\textbf{(E)} ~15{,}426$

Solution

The units digit of a multiple of $10$ will always be $0$ . We add a $0$ whenever we multiply by $10$ . So, removing the units digit is equal to dividing by $10$ .

Let the smaller number (the one we get after removing the units digit) be $a$ . This means the bigger number would be $10a$ .

We know the sum is $10a+a = 11a$ so $11a=17402$ . So $a=1582$ . The difference is $10a-a = 9a$ . So, the answer is $9(1582) = 14238 = \boxed{\textbf{(D)}}$ .

--abhinavg0627

Solution 2(Lazy Speed)

Since the ones place of a multiple of $10$ is $0$ , this implies the other integer has to end with a $2$ since both integers sum up to a number that ends with a $2$ . Thus, the ones place of the difference has to be $10-2=8$ , and the only answer choice that ends with an $8$ is $\boxed{\textbf{(D)}~14328$ (Error compiling LaTeX. Unknown error_msg)

~CoolJupiter 2021

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 (Using Algebra and Meta-solving)

https://youtu.be/d2musztzDjw

-pi_is_3.14

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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