2016 AMC 10A Problem 14

Problem 14 of 25IntermediateNumber Theory

How many ways are there to write

2016

as the sum of twos and threes, ignoring order? (For example,

1008\cdot 2 + 0\cdot 3

and

402\cdot 2 + 404\cdot 3

are two such ways.)

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Solution

The problem can be rewritten as an equation

2x + 3y = 2016,

where

x

is the number of twos and

y

is the number of threes. The goal is to find the number of multiples of

3

that can be subtracted from

2016

to result in an even number. This can be achieved by the pairs of

(1008, 0)

up to

(0, 672)

with

y

being incremented by

2.

This gives us

\dfrac{672}{2} + 1 = 337

solutions for

y

and

x.

Thus, the correct answer is C .