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

Revision as of 18:38, 22 December 2017

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 1

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 $\boxed{\textbf{(E)} \, 7}$ .

Solution 2

The sum from $a$ to $b$ where $a$ and $b$ are integers and $a>b$ is

$S=\dfrac{(a-b+1)(a+b)}{2}$

$345=\dfrac{(a-b+1)(a+b)}{2}$

$2\cdot 3\cdot 5\cdot 23=(a-b+1)(a+b)$

Let $c=a-b+1$ and $d=a+b$

$2\cdot 3\cdot 5\cdot 23=c\cdot d$

If we factor $690$ into all of its factor groups $(\text{exg}~ (10,69) ~\text{or} ~(15,46))$ we will have several ordered pairs $(c,d)$ where $c<d$

The number of possible values for $c$ is half the number of factors of $690$ which is $\frac{1}{2}\cdot2\cdot2\cdot2\cdot2=8$

However, we have one extraneous case of $(1,690)$ because here, $a=b$ and we have the sum of one consecutive number which is not allowed by the question.

Thus the answer is $8-1=7$

$\boxed{\textbf{(E)} \,7}$ .

Solution 3

There is a handy formula for this problem: The number of odd factors of $345$

$345 = 5*3*23$

$2^3 = 8$

There are 8 ways to have an increasing sum of positive integers that add to 345. However, we have to subtract one for the case where it is just $345$ . The problem wants two or more consecutive integers.

Therefore, $8-1=$ $\boxed{\textbf{(E)} \,7}$ .

Solution 4

We're dealing with an increasing arithmetic progression of common difference 1. Let $x$ be the number of terms in a summation. Let $y$ be the first term in a summation. The sum of an arithmetic progression is the average of the first term and the last term multiplied by the number of terms. The problem tells us that the sum must be 345.

$\begin{align*} x \cdot \frac{y+y+[(x-1)1]}{2}&=345 \\ 2xy+x^2-x&=690 \end{align*}$

In order to satisfy the constraints of the problem, x and y must be positive integers. Maybe we can make this into a Diophantine thing! In fact, if we just factor out that $x$ ... voilà!

$(x)(x+2y-1)=690$

There are 16 possible factor pairs to try (for brevity, I will not enumerate them here). Notice that the expression in the right parenthesis is $2y-1$ more than the expression in the parenthesis on the left. $y$ is at least 1. Thus, the expression in the right parenthesis will always be greater than the expression on the left. This eliminates 8 factor pairs. The problem also says the "increasing sequence" has to have "two or more" terms, so $x \geqslant 2$ . This eliminates the factor pair $1 \cdot 690$ . With brief testing, we find that the the other 7 factor pairs produce 7 viable ordered pairs. This means we have found $\boxed{\textbf{(E)}\ 7}$ ways to write 345 in the silly way outlined by the problem.

Solution 5

By the sum of an arithmetic sequence... this ultimately comes to $n+n+1+n+2....+n+p=345=(2n+p)(p+1)=690=23\cdot3\cdot5\cdot2$ .

Quick testing (would take you roughly a minute)

We see that the first 7values of $p$ that work are

$p=1,2,4,5,9,14,22$ .

We see that each one of them works. Hence, the answer is $\boxed{7}$ .

@@ Line 76: / Line 76: @@
 There are 16 possible factor pairs to try (for brevity, I will not enumerate them here). Notice that the expression in the right parenthesis is <math>2y-1</math> more than the expression in the parenthesis on the left. <math>y</math> is at least 1. Thus, the expression in the right parenthesis will always be greater than the expression on the left. This eliminates 8 factor pairs. The problem also says the "increasing sequence" has to have "two or more" terms, so <math>x \geqslant 2</math>. This eliminates the factor pair <math>1 \cdot 690</math>. With brief testing, we find that the the other 7 factor pairs produce 7 viable ordered pairs. This means we have found <math>\boxed{\textbf{(E)}\ 7}</math> ways to write 345 in the silly way outlined by the problem.
-==Solution 4==
+==Solution 5==
 By the sum of an arithmetic sequence... this ultimately comes to

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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