Difference between revisions of "2016 AMC 12B Problems/Problem 16"

(Solution)
(Solution)
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==Solution==
 
==Solution==
We proceed with this problem by considering two cases, when: \par
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We proceed with this problem by considering two cases, when: 1) There are an odd number of consecutive numbers, 2) There are an even number of consecutive numbers.
1. There are an odd number of consecutive numbers \par
 
2. There are an even number of consecutive numbers
 
  
 
For the first case, we can cleverly choose the convenient form of our sequence to be
 
For the first case, we can cleverly choose the convenient form of our sequence to be
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and <math>a</math> will have a solution when <math>\frac{345}{2n+1}</math> is an integer, namely when <math>2n+1</math> is a divisor of 345. We check that  
 
and <math>a</math> will have a solution when <math>\frac{345}{2n+1}</math> is an integer, namely when <math>2n+1</math> is a divisor of 345. We check that  
 
<cmath>2n+1 = 3, 5, 15, 23</cmath>
 
<cmath>2n+1 = 3, 5, 15, 23</cmath>
work, and no more, because <math>2n+1</math> does not satisfy the requirements of two or more consecutive integers, and when <math>2n+1</math> equals the next biggest factor, <math>69</math>, there must be negative integers in the sequence. Our solutions are <math>\{114,115, 116\}, \{67, \cdots, 71\}, \{16, \cdots, 30\}, \{4, \cdots, 26\}</math>.
+
work, and no more, because <math>2n+1=1</math> does not satisfy the requirements of two or more consecutive integers, and when <math>2n+1</math> equals the next biggest factor, <math>69</math>, there must be negative integers in the sequence. Our solutions are <math>\{114,115, 116\}, \{67, \cdots, 71\}, \{16, \cdots, 30\}, \{4, \cdots, 26\}</math>.
  
 
For the even cases, we choose our sequence to be of the form:
 
For the even cases, we choose our sequence to be of the form:

Revision as of 23:45, 21 February 2016

Problem

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Solution

We proceed with this problem by considering two cases, when: 1) There are an odd number of consecutive numbers, 2) There are an even number of consecutive numbers.

For the first case, we can cleverly choose the convenient form of our sequence to be \[a-n,\cdots, a-1, a, a+1, \cdots, a+n\]

because then our sum will just be $(2n+1)a$. We now have \[(2n+1)a = 345\] and $a$ will have a solution when $\frac{345}{2n+1}$ is an integer, namely when $2n+1$ is a divisor of 345. We check that \[2n+1 = 3, 5, 15, 23\] work, and no more, because $2n+1=1$ does not satisfy the requirements of two or more consecutive integers, and when $2n+1$ equals the next biggest factor, $69$, there must be negative integers in the sequence. Our solutions are $\{114,115, 116\}, \{67, \cdots, 71\}, \{16, \cdots, 30\}, \{4, \cdots, 26\}$.

For the even cases, we choose our sequence to be of the form: \[a-(n-1), \cdots, a, a+1, \cdots, a+n\] so the sum is $\frac{(2n)(2a+1)}{2} = n(2a+1)$. In this case, we find our solutions to be $\{172, 173\}, \{55,\cdots, 60\}, \{30,\cdots, 39\}$.

We have found all 7 solutions and our answer is $\boxed{\textbf{(E)} \, 7}$.

See Also

2016 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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 12 Problems and Solutions

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