2016 AMC 10A Problem 6

Problem 6 of 25EasierNumber Theory

Ximena lists the whole numbers

1

through

30

once. Emilio copies Ximena’s numbers, replacing each occurrence of the digit

2

by the digit

1.

Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena’s sum than Emilio’s?

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Solution

When Emilio changes a units digit

2

1

, his copied number is smaller by

1

. This happens in

2,12,22

, for a loss of

3

. When he changes a tens digit

2

1

, his copied number is smaller by

10

. This happens for

20,21,\ldots,29

, for a loss of

100

. Thus Emilio’s sum is

103

less than Ximena’s sum, so Ximena’s sum is

103

larger. Thus, the correct answer is D.