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2021 AMC 10A Problem 3

2021 AMC 10A Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Bases & Digits · difficulty 1 of 5 · about 2 min · position P1-10

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?

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

Answer

Key insight

Erasing a trailing zero divides by 10, so the numbers are 10n and n; their sum is 11n and their difference is 9n.

Solution

A number divisible by $10$ ends in $0$ . Erasing that final $0$ is the same as dividing by $10$ . So if the smaller number is $n$ , the larger one is $10n$ .

Their sum is $10n + n = 11n = 17{,}402$ , giving $n = 1582$ .

The difference is $10n - n = 9n = 9 \cdot 1582 = 14{,}238$ .

The answer is $\boxed{\textbf{(D)}\ 14{,}238}$ .

Why this works

Digit-erasing operations are arithmetic operations in disguise: removing a trailing zero is division by $10$ . Once both numbers are expressed with one variable, the sum and difference are simply $11n$ and $9n$ , so the answer is $\frac{9}{11}$ of the given sum.

The trap

Solving for n = 1582 and stopping, or reporting the larger number 15,820 rather than the difference.

Common mistakes

Solving for n = 1582 and stopping, or reporting the larger number 15,820 rather than the difference.
Setting the numbers as $n$ and $n + 10$ or $n$ and $n/10$ with the roles reversed.

Techniques

Set up the equation/formula and compute; no special trick needed