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2007 AMC 8 Problems/Problem 10: Difference between revisions

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== Problem ==
== Problem ==


For any positive integer <math>n</math>, <math>\boxed{n}</math> to be the sum of the positive factors of <math>n</math>.
For any positive integer <math>n</math>, define <math>\boxed{n}</math> to be the sum of the positive factors of <math>n</math>.
For example, <math>\boxed{6} = 1 + 2 + 3 + 6 = 12</math>. Find <math>\boxed{\boxed{11}}</math> .
For example, <math>\boxed{6} = 1 + 2 + 3 + 6 = 12</math>. Find <math>\boxed{\boxed{11}}</math> .


<math>\mathrm{(A)}\ 13 \qquad \mathrm{(B)}\ 20 \qquad \mathrm{(C)}\ 24 \qquad \mathrm{(D)}\ 28 \qquad \mathrm{(E)}\ 30</math>
<math>\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30</math>


== Solution ==
== Solution ==
We have
<cmath>\begin{align*}
\boxed{\boxed{11}}&=\boxed{1+11} \\
&=\boxed{12} \\
&=1+2+3+4+6+12 \\
&=28,
\end{align*}</cmath>
from which the answer is <math>\boxed{\textbf{(D)}\ 28}.</math>


First we find <math>\boxed{11}</math>.
~Aplus95 (Fundamental Logic)


<math>\boxed{11} = 1 + 11 = 12</math>
~MRENTHUSIASM (Reconstruction)
 
Then we find <math>\boxed{12}</math>.
 
<math>\boxed{\boxed{11}} = \boxed{12} = 1 + 2 + 3 + 4 + 6 + 12 = \boxed{\textbf{(D)}\ 28}</math>


==Video Solution by WhyMath==
==Video Solution by WhyMath==
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~savannahsolver
~savannahsolver
 
==Video Solution by SpreadTheMathLove==
https://www.youtube.com/watch?v=omFpSGMWhFc
==See Also==
==See Also==
{{AMC8 box|year=2007|num-b=9|num-a=11}}
{{AMC8 box|year=2007|num-b=9|num-a=11}}
{{MAA Notice}}
{{MAA Notice}}

Latest revision as of 14:54, 2 July 2024

Problem

For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$ .

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Solution

We have \begin{align*} \boxed{\boxed{11}}&=\boxed{1+11} \\ &=\boxed{12} \\ &=1+2+3+4+6+12 \\ &=28, \end{align*} from which the answer is $\boxed{\textbf{(D)}\ 28}.$

~Aplus95 (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

Video Solution by WhyMath

https://youtu.be/Ih8lEBwPqEY

~savannahsolver

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=omFpSGMWhFc

See Also

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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