2015 AMC 8 Problems/Problem 14

Which of the following integers cannot be written as the sum of four consecutive odd integers?

$\textbf{(A)}\text{ 16}\quad\textbf{(B)}\text{ 40}\quad\textbf{(C)}\text{ 72}\quad\textbf{(D)}\text{ 100}\quad\textbf{(E)}\text{ 200}$

Solution 1

Let our $4$ numbers be $n, n+2, n+4, n+6$ , where $n$ is odd. Then our sum is $4n+12$ . The only answer choice that cannot be written as $4n+12$ , where $n$ is odd, is $\boxed{\textbf{(D)}\text{ 100}}$ .

Solution 2

If the four consecutive odd integers are $2n-3,~ 2n-1, ~2n+1$ and $2n+3$ then the sum is $8n$ . All the integers are divisible by $8$ except $100$ , so the answer is $\boxed{\textbf{(D)}~100}$ .

2015 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2015 AMC 8 Problems/Problem 14

Solution 1

Solution 2

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