Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2016 AMC 10A Problems/Problem 14"

(→‎Solution 2)
Line 7: Line 7:
 
The amount of twos in our sum ranges from <math>0</math> to <math>1008</math>, with differences of <math>3</math> because <math>2 \cdot 3 = lcm(2, 3)</math>.
 
The amount of twos in our sum ranges from <math>0</math> to <math>1008</math>, with differences of <math>3</math> because <math>2 \cdot 3 = lcm(2, 3)</math>.
  
−
The possible amount of twos is <math>\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}</math>
+
The possible amount of twos is <math>\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}</math>.
  
 
==Solution 2==
 
==Solution 2==
  
−
You can also see that you can rewrite the word problem into a equation <math>2x</math> + <math>3y</math> = <math>2016</math>. Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if <math>y</math> = 0, <math>x</math> = 1008. All the way to <math>x</math> = 0, and <math>y</math> = <math>672</math>.Therefore, between <math>0</math> and <math>672</math>, the number of multiples of 2 is \boxed{\textbf{(C)}\$ 337}$.
+
You can also see that you can rewrite the word problem into a equation <math>2x</math> + <math>3y</math> = <math>2016</math>. Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if <math>y = 0</math>, <math>x</math> = 1008. All the way to <math>x = 0</math>, and <math>y = 672</math>.Therefore, between <math>0</math> and <math>672</math>, the number of multiples of 2 is <math>\boxed{\textbf{(C)}337}</math>.

Revision as of 19:22, 3 February 2016

Problem

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.)

$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

Solution 1

The amount of twos in our sum ranges from $0$ to $1008$, with differences of $3$ because $2 \cdot 3 = lcm(2, 3)$.

The possible amount of twos is $\frac{1008 - 0}{3} + 1 \Rightarrow \boxed{\textbf{(C)} 337}$.

Solution 2

You can also see that you can rewrite the word problem into a equation $2x$ + $3y$ = $2016$. Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if $y = 0$, $x$ = 1008. All the way to $x = 0$, and $y = 672$.Therefore, between $0$ and $672$, the number of multiples of 2 is $\boxed{\textbf{(C)}337}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_10A_Problems/Problem_14&oldid=75162"