Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1996 AHSME Problems/Problem 29"

(→‎Solution 2)
(→‎Solution 2)
Line 37: Line 37:
 
In this case, <math>6n = 3^4\cdot 2^6</math>, which has <math>5\cdot 7 = 35</math> factors, and the answer is <math>\boxed{C}</math>
 
In this case, <math>6n = 3^4\cdot 2^6</math>, which has <math>5\cdot 7 = 35</math> factors, and the answer is <math>\boxed{C}</math>
  
−
 
−
== Solution 2 ==
 
−
Because <math>2n</math> has <math>28</math> factors and <math>3n</math> has <math>30</math> factors, we should rewrite the number <math>n = 2^{e_1}3^{e_2}... p_n^{e_n}</math>
 
−
As the formula for the number of divisors for such a number gives:
 
−
<math>(e_1+1)(e_2+1)... (e_n+1)</math>
 
−
We plug in the variations we need to make for the cases <math>2n</math> and <math>3n</math>.
 
−
<math>2n</math> has <math>(e_1+2)(e_2+1)(e_3+1)... (e_n+1) = 28</math>
 
−
<math>3n</math> has <math>(e_1+1)(e_2+2)(e_3+1)...(e_n+1) = 30</math>
 
−
 
−
If we take the  top and divide by the bottom, we get the following equation:
 
−
<math>\frac{(e_1+2)(e_2+1)}{(e_1+1)(e_2+2)} = \frac{14}{15}</math>. Letting <math>e_1=x</math> and <math>e_2 = y</math> for convenience and expanding this out gives us:
 
−
<math>xy-13x+16y+2=0</math>
 
−
 
−
We can use Simon's Favorite Factoring Trick (SFFT) to turn this back into:
 
−
<math>(x+16)(y-13) +2 + 208 = 0</math>
 
−
<math>(x+16)(y-13) = - 210</math>
 
−
 
−
As we want to be dealing with rather reasonable numbers for <math>x</math> and <math>y</math>, we try to make the <math>x+16</math> term the slightly larger term and the <math>y-13</math> term the slightly smaller term. This effect is achieved when <math>x+16 = 21</math> and <math>y-13 = -10</math>. Therefore, <math>x = 5, y = 3</math>. We get that this already satisfies the requirements for the number we are looking for, and we take $(5+2)(3+2) = 35
 
  
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=28|num-a=30}}
 
{{AHSME box|year=1996|num-b=28|num-a=30}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:34, 6 November 2020

Problem

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have?

$\text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38$

Solution 1

Working with the second part of the problem first, we know that $3n$ has $30$ divisors. We try to find the various possible prime factorizations of $3n$ by splitting $30$ into various products of $1, 2$ or $3$ integers.

$30 \rightarrow p^{29}$

$2 \cdot 15 \rightarrow pq^{14}$

$3\cdot 10 \rightarrow p^2q^9$

$5\cdot 6 \rightarrow p^4q^5$

$2\cdot 3\cdot 5 \rightarrow pq^2r^4$

The variables $p, q, r$ are different prime factors, and one of them must be $3$. We now try to count the factors of $2n$, to see which prime factorization is correct and has $28$ factors.

In the first case, $p=3$ is the only possibility. This gives $2n = 2\cdot p^{28}$, which has $2\cdot {29}$ factors, which is way too many.

In the second case, $p=3$ gives $2n = 2q^{14}$. If $q=2$, then there are $16$ factors, while if $q\neq 2$, there are $2\cdot 15 = 30$ factors.

In the second case, $q=3$ gives $2n = 2p3^{13}$. If $p=2$, then there are $3\cdot 13$ factors, while if $p\neq 2$, there are $2\cdot 2 \cdot 13$ factors.

In the third case, $p=3$ gives $2n = 2\cdot 3\cdot q^9$. If $q=2$, then there are $11\cdot 2 = 22$ factors, while if $q \neq 2$, there are $2\cdot 2\cdot 10$ factors.

In the third case, $q=3$ gives $2n = 2\cdot p^2\cdot 3^8$. If $p=2$, then there are $4\cdot 9$ factors, while if $p \neq 2$, there are $2\cdot 3\cdot 9$ factors.

In the fourth case, $p=3$ gives $2n = 2\cdot 3^3\cdot q^5$. If $q=2$, then there are $7\cdot 4= 28$ factors. This is the factorization we want.

Thus, $3n = 3^4 \cdot 2^5$, which has $5\cdot 6 = 30$ factors, and $2n = 3^3 \cdot 2^6$, which has $4\cdot 7 = 28$ factors.

In this case, $6n = 3^4\cdot 2^6$, which has $5\cdot 7 = 35$ factors, and the answer is $\boxed{C}$


See also

1996 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 28 		Followed by
Problem 30
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1996_AHSME_Problems/Problem_29&oldid=136849"