Track / Stage 3 / 148 of 260 #148 of 1964

Problem 148

AMC 10/12, early questions

Number theory Difficulty 3.5 Find the answer

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

Your answer

Multiple choice: answer with the letter of the option you want.

Official 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 $9(1582) = \boxed{\textbf{(D)} ~14{,}238}$ .
~abhinavg0627

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.