Difference between revisions of "1987 AIME Problems/Problem 11"

Latest revision as of 17:30, 20 January 2024

Problem

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

Solutions

Solution 1

Let us write down one such sum, with $m$ terms and first term $n + 1$ :

$3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)$ .

Thus $m(2n + m + 1) = 2 \cdot 3^{11}$ so $m$ is a divisor of $2\cdot 3^{11}$ . However, because $n \geq 0$ we have $m^2 < m(m + 1) \leq 2\cdot 3^{11}$ so $m < \sqrt{2\cdot 3^{11}} < 3^6$ . Thus, we are looking for large factors of $2\cdot 3^{11}$ which are less than $3^6$ . The largest such factor is clearly $2\cdot 3^5 = 486$ ; for this value of $m$ we do indeed have the valid expression $3^{11} = 122 + 123 + \ldots + 607$ , for which $k=\boxed{486}$ .

Solution 2

First note that if $k$ is odd, and $n$ is the middle term, the sum equals $kn$ . If $k$ is even, then we have the sum equal to $kn+k/2$ , which will be even. Since $3^{11}$ is odd, we see that $k$ is odd.

Thus, we have $nk=3^{11} \implies n=3^{11}/k$ . Also, note $n-(k+1)/2=0 \implies n=(k+1)/2.$ Subsituting $n=3^{11}/k$ , we have $k^2+k=2*3^{11}$ . Proceed as in solution 1.

Solution 3

Proceed as in Solution 1 until it is noted that $m$ is a divisor of $2\cdot 3^{11}$ . The divisors of $2\cdot 3^{11}$ are $3^{1} , 2\cdot 3^{1} , 3^{2} , 2\cdot 3^{2} , \ldots , 2\cdot 3^{10} , 3^{11}$ . Note that the factors of $m(2n + m + 1)$ are of opposite parity (if $m$ is odd, then $(2n + m + 1)$ is even and vice versa). Thus, one of the two factors will be a power of three, and the other will be twice a power of three. $(2n + m + 1)$ will represent the greater factor while $m$ will represent the lesser factor. Given this information, we need to find the factor pair that maximizes the lesser of the two factors, as this will maximize the value of $m$ . The factor pair which maximizes the lesser factor is $2\cdot 3^{5}$ and $3^{6}$ . It follows that $m$ = $2\cdot 3^{5}$ = $\boxed{486}$ .

~ cxsmi

@@ Line 17: / Line 17: @@
 Proceed as in Solution 1 until it is noted that <math>m</math> is a divisor of <math>2\cdot 3^{11}</math>. The divisors of <math>2\cdot 3^{11}</math> are <math>3^{1} , 2\cdot 3^{1} , 3^{2} , 2\cdot 3^{2} , \ldots , 2\cdot 3^{10} , 3^{11}</math>. Note that the factors of <math>m(2n + m + 1)</math> are of opposite parity (if <math>m</math> is odd, then <math>(2n + m + 1)</math> is even and vice versa). Thus, one of the two factors will be a power of three, and the other will be twice a power of three. <math>(2n + m + 1)</math> will represent the greater factor while <math>m</math> will represent the lesser factor. Given this information, we need to find the factor pair that maximizes the lesser of the two factors, as this will maximize the value of <math>m</math>. The factor pair which maximizes the lesser factor is <math>2\cdot 3^{5}</math> and <math>3^{6}</math>. It follows that <math>m</math> = <math>2\cdot 3^{5}</math> = <math>\boxed{486}</math>.
-~ cxsmi
+~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
 == See also ==

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search