2021 AMC 10A Spring Problem 3

Question

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?

10{,}272

11{,}700

13{,}362

14{,}238

15{,}426

If deleting the final zero gives the other number, the larger number is ten times the smaller one

Use the sum to find the smaller number first

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Let

x

and

y

be the two numbers. WLOG, let

x

be divisible by

10.

Then the units digit of

x

is

0.

If we erase the units digit, then we are essentially dividing

x

by

10.

The problem statement also gives us that

\dfrac{x}{10} = y.

Therefore,

\begin{aligned} \dfrac{x}{10} + x &= 17,402 \\ \dfrac{11x}{10} &= 17,402 \\ x &= 15,820. \end{aligned}

Then

x - \dfrac{x}{10} = 14,238.

Thus, D is the correct answer.

Problem 3 in other years