2021 AMC 12A 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

Erasing the units digit

0

of a multiple of

10

divides it by

10

If the smaller number is

x,

the larger is

10x,

so

11x = 17{,}402

The larger number ends in

0,

and erasing that digit divides it by

10

to give the smaller number. So the larger number is

10

times the smaller. Writing the smaller number as

x,

the sum is

x + 10x = 11x = 17{,}402,

giving

x = 1{,}582.

The two numbers are

1{,}582

and

15{,}820,

whose difference is

15{,}820 - 1{,}582 = 14{,}238.

Thus, the correct answer is D.

Problem 3 in other years