2016 AMC 10A Problem 14

Question

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.)

236

336

337

403

672

Write

2x+3y=2016

y

must be even from

0

through

672

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 .

Problem 14 in other years