HMMT November, 2014

Algebra Difficulty 4.0 AIME Find the answer United States

Problem:
How many integers $n$ in the set $\{4,9,14,19, \ldots, 2014\}$ have the property that the sum of the decimal digits of $n$ is even?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Answer: 201
We know that 2014 does not qualify the property. So, we'll consider $\{4,9,14, \ldots, 2009\}$ instead. Now, we partition this set into 2 sets: $\{4,14,24, \ldots, 2004\}$ and $\{9,19,29, \ldots, 2009\}$ .

For each, the first and second set are basically $x4$ and $x9$ , where $x = 0, 1, 2, \ldots, 200$ , respectively. And we know that for each value of $x$ , $x$ must be either even or odd, which makes exactly one of $\{x4, x9\}$ have even sum of decimal digits. Therefore, there are in total 201 such numbers.

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