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Difference between revisions of "2013 AMC 10B Problems/Problem 24"

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<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5</math>
 
<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5</math>
  
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==Solution 1==
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==Solution==
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A positive integer with only four positive divisors has its prime factorization in the form of <math>a*b</math>, where <math>a</math> and <math>b</math> are both prime positive integers or <math>c^3</math> where <math>c</math> is a prime. One can easily deduce that none of the numbers are even near a cube so that case is finished. We now look at the case of <math>a*b</math>. The four factors of this number would be <math>1</math>, <math>a</math>, <math>b</math>, and <math>ab</math>.  The sum of these would be <math>ab+a+b+1</math>, which can be factored into the form <math>(a+1)(b+1)</math>. Easily we can see that now we can take cases again.  
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A positive integer with only four positive divisors has its prime factorization in the form of <math>a \cdot b</math>, where <math>a</math> and <math>b</math> are both prime positive integers or <math>c^3</math> where <math>c</math> is a prime. One can easily deduce that none of the numbers are even near a cube so the second case is not possible. We now look at the case of <math>a \cdot b</math>. The four factors of this number would be <math>1</math>, <math>a</math>, <math>b</math>, and <math>ab</math>.  The sum of these would be <math>ab+a+b+1</math>, which can be factored into the form <math>(a+1)(b+1)</math>. Easily we can see that now we can take cases again.  
  
 
Case 1: Either <math>a</math> or <math>b</math> is 2.  
 
Case 1: Either <math>a</math> or <math>b</math> is 2.  
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This implies that both <math>(a+1)</math> and <math>(b+1)</math> are even which implies that in this case the number must be divisible by <math>4</math>. This leaves only <math>2012</math> and <math>2016</math>.
 
This implies that both <math>(a+1)</math> and <math>(b+1)</math> are even which implies that in this case the number must be divisible by <math>4</math>. This leaves only <math>2012</math> and <math>2016</math>.
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<math>2012=4*503</math> so either <math>(a+1)</math> or <math>(b+1)</math> both has a factor of <math>2</math> or one has a factor of <math>4</math>. If it was the first case, then <math>a</math> or <math>b</math> will equal <math>1</math>.  That means that either <math>(a+1)</math> or <math>(b+1)</math> has a factor of <math>4</math>.  That means that <math>a</math> or <math>b</math> is <math>502</math> which isn't a prime, so 2012 does not work.  <math>2016 = 4 * 504</math> so we have <math>(503 + 1)(3 + 1)</math>. 503 and 3 are both odd primes, so 2016 is a solution. Thus the answer is <math>\boxed{\textbf{(A)}\ 1}</math>.
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<math>2012={4}\cdot{503}</math> so either <math>(a+1)</math> or <math>(b+1)</math> both have a factor of <math>2</math> or one has a factor of <math>4</math> since <math>503</math> is prime. If it was the first case, then <math>a</math> or <math>b</math> will equal <math>1</math>.  That means that either <math>(a+1)</math> or <math>(b+1)</math> has a factor of <math>4</math>.  That means that <math>a</math> or <math>b</math> is <math>502</math> which isn't a prime, so 2012 does not work.  <math>2016 = 4 \cdot 504</math> so we have <math>(503 + 1)(3 + 1)</math>. 503 and 3 are both odd primes, so 2016 is a solution. Thus the answer is <math>\boxed{\textbf{(A)}\ 1}</math>.
  
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==Solution 2==
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===Shortcut===
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If <math>m</math> has four divisors, then its divisors would be 1, <math>a</math>, <math>b</math> and <math>ab</math>, where <math>a</math> and <math>b</math> are prime. Therefore, the sum of the divisors of <math>m</math> is <math>1+a+b+ab=(a+1)(b+1)</math>.  
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After deducing that <math>2012</math> and case <math>1</math> is impossible, and since there is no option for <math>0</math>, <math>2016</math> is obviously a solution and the answer is <math>\boxed{\textbf{(A)}\ 1}</math>.
  
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If either <math>a+1</math> or <math>b+1</math> are odd, then <math>a</math> or <math>b</math> are even. Therefore, <math>a+1</math> and <math>b+1</math> are even, so <math>m</math> is a multiple of 4. The only two numbers from the range <math>2010-2019</math> that are multiples of 4 are 2012 and 2016.
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-mathboy282
  
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Factoring 2012, we get <math>2^2*503</math>. To make <math>a+1</math> and <math>b+1</math> even, <math>wlog</math>, we have <math>a+1=2</math> and <math>b+1=1006</math>. However, if <math>a</math> was 1, then <math>a</math> is not prime, so 2012 is not nice.
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==Video Solution==
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https://youtu.be/7QNI7VMFkow
  
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Factoring 2016, we get <math>2^5*3^2*7</math>. WLOG, we have <math>a<b</math>.
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~savannahsolver
  
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Testing for the lowest <math>a</math>, we get <math>a+1=4</math> and <math>b+1=504</math>. Therefore, <math>a=3</math>, and <math>b=503</math>, so <math>n=2016</math> is nice, with <math>m=1509</math>. Therefore, the answer is <math>\boxed{\textbf{(A)}\ 1}</math>.
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==Video Solution 2 by SpreadTheMathLove==
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https://www.youtube.com/watch?v=WzSddwBAWtA
  
 
== See also ==
 
== See also ==

Revision as of 21:55, 23 July 2024

Problem

A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numbers in the set $\{ 2010,2011,2012,\dotsc,2019 \}$ are nice?


$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

Solution

A positive integer with only four positive divisors has its prime factorization in the form of $a \cdot b$, where $a$ and $b$ are both prime positive integers or $c^3$ where $c$ is a prime. One can easily deduce that none of the numbers are even near a cube so the second case is not possible. We now look at the case of $a \cdot b$. The four factors of this number would be $1$, $a$, $b$, and $ab$. The sum of these would be $ab+a+b+1$, which can be factored into the form $(a+1)(b+1)$. Easily we can see that now we can take cases again.

Case 1: Either $a$ or $b$ is 2.

If this is true then we have to have that one of $(a+1)$ or $(b+1)$ is odd and that one is 3. The other is still even. So we have that in this case the only numbers that work are even multiples of 3 which are 2010 and 2016. So we just have to check if either $\frac{2016}{3} - 1$ or $\frac{2010}{3} - 1$ is a prime. We see that in this case none of them work.

Case 2: Both $a$ and $b$ are odd primes.

This implies that both $(a+1)$ and $(b+1)$ are even which implies that in this case the number must be divisible by $4$. This leaves only $2012$ and $2016$. $2012={4}\cdot{503}$ so either $(a+1)$ or $(b+1)$ both have a factor of $2$ or one has a factor of $4$ since $503$ is prime. If it was the first case, then $a$ or $b$ will equal $1$. That means that either $(a+1)$ or $(b+1)$ has a factor of $4$. That means that $a$ or $b$ is $502$ which isn't a prime, so 2012 does not work. $2016 = 4 \cdot 504$ so we have $(503 + 1)(3 + 1)$. 503 and 3 are both odd primes, so 2016 is a solution. Thus the answer is $\boxed{\textbf{(A)}\ 1}$.

Shortcut

After deducing that $2012$ and case $1$ is impossible, and since there is no option for $0$, $2016$ is obviously a solution and the answer is $\boxed{\textbf{(A)}\ 1}$.

-mathboy282

Video Solution

https://youtu.be/7QNI7VMFkow

~savannahsolver

Video Solution 2 by SpreadTheMathLove

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

See also

2013 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 AMC 10 Problems and Solutions

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