+0  
 
+1
77
2
avatar+187 

Given that n>1, what is the smallest positive integer n whose positive divisors have a product of n^6?

MathCuber  Nov 15, 2018
 #1
avatar
0

n=60
Total = 12 Divisors
(1* 2* 3* 4* 5* 6*10* 12* 15* 20* 30* 60)=46,656,000,000
Log(46,656,000,000) / Log(60)=60^6          =46,656,000,000    

Guest Nov 15, 2018
 #2
avatar+20680 
+11

Given that n>1, what is the smallest positive integer n whose positive divisors have a product of n^6?

 

\(\text{If $n$ is not a perfect square, the divisors of n are divided into couples.} \\ \text{For example, the divisors of $12$ are $1, 2, 3, 4, 6,$ and $12$.} \\ \text{$d(12)$ is $6$, and $1·2·3·4·6·12=1728=123=1262=12^{\frac{d(n)}{2}}$} \\ \text{In our example, $1$ and $12$ are partners, as are $2$ and $6$, as are $3$ and $4$.} \\ \text{Note that the product of any $2$ partnered numbers is $n$.} \\ \text{Then there are \frac{d(n)}{2} couples. } \\ \text{The product of the elements in any couple is $n$, } \\ \text{so the product of all the divisors of $n$ is $ n^{ \frac{d(n)}{2}}$.}\)

 

\(\text{The divisors have a product of $n^6$, so $n^6=n^{ \frac{d(n)}{2}}$, or $6 = \frac{d(n)}{2}$ } \\ \text{So we see $\mathbf{d(n) = 12}$. Our number $n$ must have $\mathbf{12}$ divisors! }\)

 

\(\text{The factorisation of $12$ is ${\color{red}3}\cdot {\color{red}2}\cdot {\color{red}2}$} \\ \text{$d(n) = (\underbrace{{\color{blue}2}+1}_{={\color{red}3}}) (\underbrace{{\color{blue}1}+1}_{={\color{red}2}}) (\underbrace{{\color{blue}1}+1}_{={\color{red}2}}) =12, $} \\ \text{so the factorisation of $ n = p_1^{\color{blue}2}p_2^{ \color{blue}1 }p_3^{ \color{blue}1 } $ } \\ \text{The smallest prime numbers are $p_1=2,~p_2=3$ and $p_3=5$ } \\ \text{So the number $n = 2^{\color{blue}2}3^{ \color{blue}1 }5^{ \color{blue}1 } = \mathbf{60} $ }\)

 

laugh

heureka  Nov 15, 2018
edited by heureka  Nov 15, 2018

6 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.