2013 AMC 10B Problem 18

Question

The number

2013

has the property that its units digit is the sum of its other digits, that is

2+0+1=3.

How many integers less than

2013

but greater than

1000

have this property?

33

34

45

46

58

Count numbers from

1001

to

1999

by their units digit

Handle the small range from

2000

to

2012

separately

Once the first three digits are given, their sum determines the units digit, provided that the sum is at most

9.

We separate the possibilities according to the thousands digit.

If the thousands digit is

1,

then the sum of the hundreds and tens digits must be at most

8.

For each possible sum

s,

there are

s+1

choices, because the hundreds digit can range from

0

to

s.

Thus, this case gives the ninth triangular number:

\dfrac{9\cdot 10}2 =45.

If the thousands digit is

2,

the only qualifying number below

2013

is

2002,

giving

46

cases in total.

Thus, the correct answer is D.

Problem 18 in other years