# 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.

## 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 1 (Algebra)

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 $9(1582) = \boxed{\textbf{(D)} ~14{,}238}$.

~abhinavg0627

## Solution 2 (Arithmetic)

Since the unit's place of a multiple of $10$ is $0$, the other integer must end with a $2$, for both integers sum up to a number ending in a $2$. Thus, the unit's place of the difference must be $10-2=8$, and the only answer choice that ends with an $8$ is $\boxed{\textbf{(D)} ~14{,}238}$.

Another quick solution is to realize that the sum represents a number $n$ added to $10n$. The difference is $9n$, which is $\frac{9}{11}$ of the given sum.

~CoolJupiter 2021

~Extremelysupercooldude (Minor grammar edits)

## Solution 3 (Vertical Addition and Logic)

Let the larger number be $\underline{AB{,}CD0}.$ It follows that the smaller number is $\underline{A{,}BCD}.$ Adding vertically, we have $$\begin{array}{cccccc} & A & B & C & D & 0 \\ +\quad & & A & B & C & D \\ \hline & & & & & \\ [-2.5ex] & 1 & 7 & 4 & 0 & 2 \\ \end{array}$$ Working from right to left, we get $$D=2\implies C=8 \implies B=5 \implies A=1.$$ The larger number is $15{,}820$ and the smaller number is $1{,}582.$ Their difference is $15{,}820-1{,}582=\boxed{\textbf{(D)} ~14{,}238}.$

~MRENTHUSIASM

## Solution 4 (Logic)

We know that the larger number has a units digit of $0$ since it is divisible by 10. If $D$ is the ten's digit of the larger number, then $D$ is the units digit of the smaller number. Since the sum of the natural numbers has a unit's digit of $2$, $D=2$.

The units digit of the larger number is $0$ and the units digit of the smaller number is $2$, so the positive difference between the numbers is 8. There is only one answer choice that has this units digit, and that is $\boxed{\textbf{(D)} ~14{,}238}.$

(Similar to MRENTHUSIASM's solution)

## Video Solution 1 (easily understandable)

### Video Solution (Simple)

~ Education, the study Of Everything

-pi_is_3.14

~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

~MathWithPi