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Difference between revisions of "2005 AMC 10A Problems/Problem 21"

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==Problem==
 
==Problem==
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For how many positive integers <math>n</math> does <math> 1+2+...+n </math> evenly divide from <math>6n</math>?  
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For how many positive integers <math>n</math> does <math>1+2+\dotsb+n</math> evenly divide <math>6n</math>?  
  
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<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 11 </math>
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<math>
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\textbf{(A) } 3\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 9\qquad \textbf{(E) } 11
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</math>
  
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==Solution 1==
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== Solution ==
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If <math> 1+2+...+n </math> evenly [[divide]]s <math>6n</math>, then <math>\frac{6n}{1+2+...+n}</math> is an [[integer]].  
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By a standard result, <math>1+2+\dotsb+n = \frac{n(n+1)}{2}</math>, so this will evenly divide <math>6n</math> precisely if <math>\frac{6n}{\left(\frac{n(n+1)}{2}\right)} = \frac{12}{n+1}</math> is an integer, or equivalently, <math>(n+1)</math> is a factor of <math>12</math>.  
  
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Since <math> 1+2+...+n = \frac{n(n+1)}{2} </math> we may substitute the [[RHS]] in the above [[fraction]].  
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The positive factors of <math>12</math> are <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>6</math>, and <math>12</math> (i.e. <math>6</math> factors in total), but as <math>n</math> must be a positive integer, we must have <math>n+1 \geq 1+1 = 2</math>. Therefore, the factor <math>1</math> cannot be used, but all the others can, giving <math>6-1 = \boxed{\textbf{(B) } 5}</math> possible values of <math>n</math> (namely, <math>n</math> can be <math>2-1 = 1</math>, <math>3-1 = 2</math>, <math>4-1 = 3</math>, <math>6-1 = 5</math>, or <math>12-1 = 11</math>).
  
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So the problem asks us for how many [[positive integer]]s <math>n</math> is <math>\frac{6n}{\frac{n(n+1)}{2}}=\frac{12}{n+1}</math> an integer.  
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==Video Solution==
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https://youtu.be/WRv86DHa3zY
  
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<math>\frac{12}{n+1}</math> is an integer when <math>n+1</math> is a [[divisor |factor]] of <math>12</math>.
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==See also==
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{{AMC10 box|year=2005|ab=A|num-b=20|num-a=22}}
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The factors of <math>12</math> are <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>6</math>, and <math>12</math>.
 
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So the possible values of <math>n</math> are <math>0</math>, <math>1</math>, <math>2</math>, <math>3</math>, <math>5</math>, and <math>11</math>.
 
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But <math>0</math> isn't a positive integer, so only <math>1</math>, <math>2</math>, <math>3</math>, <math>5</math>, and <math>11</math> are possible values of <math>n</math>.
 
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Therefore the number of possible values of <math>n</math> is <math>5\Longrightarrow \mathrm{(B)}</math>
 
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== Solution 2 ==
 
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The sum of the first <math>n</math> positive integers is <math>\frac{n(n+1)}{2}</math>. If this is to divide <math>6n</math>, then there exists a positive integer <math>k</math> such that:
 
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<math>k \cdot \frac{n(n+1)}{2} = 6n</math>
 
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<math>k(n+1) = 12</math>
 
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Therefore, <math>k</math> and <math>n+1</math> are divisors of <math>12</math>. There are <math>6</math> divisors of <math>12</math>, which are <math>1, 2, 3, 4, 6, 12</math>. The divisors which multiply to <math>12</math> can be assigned to <math>k</math> and <math>n+1</math> in either order. However, when <math>1</math> is assigned to <math>n+1</math>, then <math>n=0</math>, which is not possible, because <math>n</math> must be positive. Therefore, we have <math>6-1=5</math> values of <math>n</math> <math>\Rightarrow \boxed{B}</math>
 
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==See Also==
 
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*[[2005 AMC 10A Problems]]
 
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*[[2005 AMC 10A Problems/Problem 20|Previous Problem]]
 
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*[[2005 AMC 10A Problems/Problem 22|Next Problem]]
 
  
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]
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{{MAA Notice}}

Latest revision as of 02:09, 2 July 2025

Problem

For how many positive integers $n$ does $1+2+\dotsb+n$ evenly divide $6n$?

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

Solution

By a standard result, $1+2+\dotsb+n = \frac{n(n+1)}{2}$, so this will evenly divide $6n$ precisely if $\frac{6n}{\left(\frac{n(n+1)}{2}\right)} = \frac{12}{n+1}$ is an integer, or equivalently, $(n+1)$ is a factor of $12$.

The positive factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$ (i.e. $6$ factors in total), but as $n$ must be a positive integer, we must have $n+1 \geq 1+1 = 2$. Therefore, the factor $1$ cannot be used, but all the others can, giving $6-1 = \boxed{\textbf{(B) } 5}$ possible values of $n$ (namely, $n$ can be $2-1 = 1$, $3-1 = 2$, $4-1 = 3$, $6-1 = 5$, or $12-1 = 11$).

Video Solution

https://youtu.be/WRv86DHa3zY

See also

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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All AMC 10 Problems and Solutions

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