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?

$33$

$34$

$45$

$46$

$58$

Answer: D

Concepts:digits casework triangular number

Difficulty rating: 1420

Small Hint:

Count numbers from $1001$ to $1999$ by their units digit

Big Hint:

Handle the small range from $2000$ to $2012$ separately

Solution:

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.

2013 AMC 10B Problem 18

Answer: D

Difficulty rating: 1420

Small Hint:

Big Hint:

Solution:

Problem 18 in Other Years