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2005 AMC 10A Problems/Problem 21

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Problem

For how many positive integers $n$ does $1+2+...+n$ evenly divide from $6n$ ?

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 11$

Solution

If $1+2+...+n$ evenly divides $6n$ , then $\frac{6n}{1+2+...+n}$ is an integer.

Since $1+2+...+n = \frac{n(n+1)}{2}$ we may substitute the RHS in the above fraction. So the problem asks us for how many positive integers $n$ is $\frac{6n}{\frac{n(n+1)}{2}}=\frac{12}{n+1}$ an integer, or equivalently when $k(n+1) = 12$ for a positive integer $k$ .

$\frac{12}{n+1}$ is an integer when $n+1$ is a factor of $12$ .

The factors of $12$ are $1$ , $2$ , $3$ , $4$ , $6$ , and $12$ , so the possible values of $n$ are $0$ , $1$ , $2$ , $3$ , $5$ , and $11$ .

But $0$ isn't a positive integer, so only $1$ , $2$ , $3$ , $5$ , and $11$ are possible values of $n$ . Therefore the number of possible values of $n$ is $5\Longrightarrow \boxed{\mathrm{(B)}}$ .

See also

2005 AMC 10A (Problems • Resources)
Preceded by Problem 20	Followed by Problem 22
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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2005_AMC_10A_Problems/Problem_21"

Category: Introductory Number Theory Problems

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