AMC 10 question

Question

Question

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

---

Answer

Factor $345=3 \times 5 \times 23$.

Suppose we take an odd number $k$ of consecutive integers, centered on $m$. Then $mk=345$ with $\frac12k<m$. Looking at the factors of $345$, the possible values of $k$ are $3,5,15,23$ centred on $115,69,23,15$ respectively.

Suppose instead we take an even number $2k$ of consecutive integers, centred on $m$ and $m+1$. Then $k(2m+1)=345$ with $k\le m$. Looking again at the factors of $345$, the possible values of $k$ are $1,3,5$ centered on $(172,173),(57,58),(34,35)$ respectively.

Thus the answer is $\textbf{(E) }7$.

Can someone explain to me how they got $0.5 k < m$ and the part saying centered on $m$ and $m + 1$.

Answer 1

Observation: If you add an odd collection of $k$ consecutive integers, you get a multiple of the middle number. In other words, if you add $$ (m-r)+(m-r+1)+\cdots+(m-2)+(m-1)+m+(m+1)+(m+2)+\cdots+(m+r), $$ there is a lot of cancellation, and you get $m(2r+1)$ as the sum, since there are $(2r+1)$ terms in the sum. To avoid nonpositive numbers, we need the smallest element in this list to be positive, so that $m-r>0$ or that $m>r$.

In your case, the total number of summands is $k=2r+1$ so $r=\frac{k-1}{2}$. Plugging this into the inequality above, you get: $$ m>\frac{k-1}{2}. $$ The given inequality of $$ m+\frac{1}{2}k $$ implies this necessary inequality.

Observation: If you add an even collection of $k$ integers, you get a multiple of the average of the middle two numbers. In other words, if you add $$ (m-r)+\cdots+(m-2)+(m-1)+m+(m+1)+((m+1)+1)+((m+1)+2)+\cdots+((m+1)+r), $$ then you can pair up the terms to get $r+1$ copies of $m+(m+1)$. Therefore, this sum is $(2m+1)(r+1)$.

Once again, you need the smallest number in this list to be positive, so $m-r>0$ or that $m>r$. My $r$ differs from your $k$ by $1$, so the strict inequality becomes $\geq$ in your case.

Answer 2

Centered on $m$ and $m+1$ just means centered between those two numbers, i.e. $\frac{2m+1}{2}$. When $k=2$, your list is $\{m,m+1\}$, for $k=4$ your list is $\{m-1,m,m+1,m+2\}$, and so on.

For odd $k$, your list is centered on an integer $m$. When $k=3$, your list is $\{m-1,m,m+1\}$, for $k=5$ your list is $\{m-2,m-1,m,m+1,m+2\}$, and so on.

When summing k consecutive integers centered on $m$, your total $Y=km$. In your case, $Y=345$ and $m=3,5,15,$ or $23$ since those are the factors of 345.