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.)

$236$

$336$

$337$

$403$

$672$

Answer: C

Concepts:Diophantine Equation parity

Difficulty rating: 1280

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 .

2016 AMC 10A Problem 14

Answer: C

Difficulty rating: 1280

Solution:

Problem 14 in Other Years