Difference between revisions of "2016 AMC 10A Problems/Problem 14"
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<math>\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672</math> | <math>\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672</math> | ||
| − | ==Solution== | + | ==Solution 1 == |
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== | ||
| + | |||
| + | 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 \makebox[2 in]{(C) 337} | ||
Revision as of 18:41, 3 February 2016
Problem
How many ways are there to write 
as the sum of twos and threes, ignoring order? (For example, 
and 
are two such ways.)
Solution 1
The amount of twos in our sum ranges from 
to 
, with differences of 
because 
.
The possible amount of twos is 
Solution 2
You can also see that you can rewrite the word problem into a equation 
+ 
= . Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if 
= 0, 
= 1008. All the way to = 0, and = 
.Therefore, between and , the number of multiples of 2 is \makebox[2 in]{(C) 337}
