# 2006 AMC 12A Problems/Problem 8

The following problem is from both the 2006 AMC 12A #8 and 2008 AMC 10A #9, so both problems redirect to this page.

## Problem

How many sets of two or more consecutive positive integers have a sum of $15$?

$\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

## Solution 1

Notice that if the consecutive positive integers have a sum of $15$, then their average (which could be a fraction) must be a divisor of $15$. If the number of integers in the list is odd, then the average must be either $1, 3,$ or $5$, and $1$ is clearly not possible. The other two possibilities both work:

• $1 + 2 + 3 + 4 + 5 = 15$
• $4 + 5 + 6 = 15$

If the number of integers in the list is even, then the average will have a $\frac{1}{2}$. The only possibility is $\frac{15}{2}$, from which we get:

• $15 = 7 + 8$

Thus, the correct answer is $\boxed{\textbf{(C) }3}.$

Question: (RealityWrites) Is it possible that the answer is $4$, because $0+1+2+3+4+5$ should technically count, right?

Answer: (IMGROOT2) It isn't possible because the question asks for positive integers, and this means that negative integers or zero aren't allowed.

Note to readers: make sure to always read the problem VERY carefully before attempting; it could mean the difference of making the cutoff.

## Solution 2

Any set will form a arithmetic progression with the first term say $a$. Since the numbers are consecutive the common difference $d = 1$.

The sum of the AP has to be 15. So,

$$S_n = \frac{n}{2} \cdot (2a + (n-1)d)$$ $$S_n = \frac{n}{2} \cdot (2a + (n-1)1)$$ $$15 = \frac{n}{2} \cdot (2a + n - 1)$$ $$2an + n^2 - n = 30$$ $$n^2 + n(2a - 1) - 30 = 0$$

Now we need to find the number of possible sets of values of a, n which satisfy this equation. Now $a$ cannot be 15 as we need 2 terms. So a can only be less the 15.

Trying all the values of a from 1 to 14 we observe that $a = 4$, $a = 7$ and $a = 1$ provide the only real solutions to the above equation.The three possibilites of a and n are.

$$(a,n) = (4,3),(7, 2),(1, 6)$$

The above values are obtained by solving the following equations obtained by substituting the above mentioned values of a into $n^2 + n(2a - 1) - 30 = 0$,

$$n^2 + 7n - 30 = 0$$ $$n^2 + 13n - 30 = 0$$ $$n^2 - n - 30 = 0$$

Since there are 3 possibilities the answer is $\boxed{\textbf{(C) }3}.$