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

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==Solution 2==
 
==Solution 2==
  
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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 2016 will be an even number. We can see if <math>y = 0</math>, <math>x = 1008</math>. 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>.
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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 <math>3</math> subtracted from 2016 will be an even number. We can see that <math>x = 1008</math>, <math>y = 0</math> all the way to <math>x = 0</math>, and <math>y = 672</math> works, with <math>y</math> being incremented by <math>2</math>'s.Therefore, between <math>0</math> and <math>672</math>, the number of multiples of <math>2</math> is <math>\boxed{\textbf{(C)}337}</math>.
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==See Also==
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{{AMC10 box|year=2016|ab=A|num-b=13|num-a=15}}
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{{MAA Notice}}

Revision as of 21:52, 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 2016 will be an even number. We can see that $x = 1008$, $y = 0$ all the way to $x = 0$, and $y = 672$ works, with $y$ being incremented by $2$'s.Therefore, between $0$ and $672$, the number of multiples of $2$ is $\boxed{\textbf{(C)}337}$.

See Also

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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