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I am doing a review assignment and I'm stuck on this problem.

a) How many positive divisors does $a$ have? I got $60$

b) How many positive integers less than $a$ are relatively prime to $a$? I got $720$

c) What is the smallest positive integer $m$ such that $a^2m$ is a cube?

d) list all positive divisors $b$ of a for which a divides $b^2$ is also true.

Any help and advice would be greatly appreciated. Thank you for your time and help!

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    $\begingroup$ I would suggest writing an algorithm (in something like Python) which will spit out all of this information if you plug in any number for $a$. $\endgroup$ – Forever Mozart Dec 10 '14 at 19:55
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First, note that $43120 = 2^\color{red}{4}\cdot 5^\color{blue}{1}\cdot 7^\color{green}{2}\cdot 11^\color{purple}{1}$.

Part A

The number of positive integer divisors is the product of one plus each exponent in the prime factorization. That is, $$d(43120) = \color{red}{(4+1)}\color{blue}{(1+1)}\color{green}{(2+1)}\color{purple}{(1+1)} = (5)(2)(3)(2) = 60$$

Part B

The number of positive integers coprime to $43120$ can be found using Euler's Totient function, $\phi(n)$. Since $\phi$ is multiplicative for coprime integers:

$$\phi(43120) = \phi(2^4)\phi(5)\phi(7^2)\phi(11)$$

Also, $\phi(p^n) = p^{n-1}(p-1)$ for prime integers $p$. Then:

$$\begin{align}\phi(43120) &= \left[2^3(2-1)\right]\left[5-1\right]\left[7^1(7-1)\right]\left[11-1\right]\\ &= 8\cdot4\cdot 42\cdot 10\\ &= 13440 \end{align}$$

Part C We want all exponents of $a^2$ to be multiples of three, and we want the smallest such exponents. $$\left(2^\color{red}{4}\cdot 5^\color{blue}{1}\cdot 7^\color{green}{2}\cdot 11^\color{purple}{1}\right)^2 = 2^\color{red}{8}\cdot 5^\color{blue}{2}\cdot 7^\color{green}{4}\cdot 11^\color{purple}{2}$$ Well, it is easy to see that we need to add $1$ to the first exponent, $1$ to the second, $2$ to the third and $1$ to the fourth. Thus, our integer $m$ is: $$m= 2^\color{red}{1}\cdot 5^\color{blue}{1}\cdot 7^\color{green}{2}\cdot 11^\color{purple}{1}$$

Part D This is a similar exponent-related trick as in part c if I'm thinking it through correctly. I'll leave this one as an exercise to the reader. :)

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  • $\begingroup$ You are awesome! Thank you, it was much easier with your explanations. $\endgroup$ – Murray Dec 10 '14 at 20:24
  • $\begingroup$ The answer for item d is $2^a\cdot 5\cdot 7^b\cdot 11$, with $a=2,3$ or $4$ and $c=1$ or $2$ (and so there are six such numbers)? $\endgroup$ – Larara Dec 11 '14 at 3:49
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First, we factor $$43120=2^45^17^211^1$$

Solutions to a, c, d, are obtained from the exponents $u=(4,1,2,1)$.

a) A positive divisor is a $4$-vector of nonnegative integers majorized by $u$. That is $$\{(a,b,c,d):0\le a\le 4, 0\le b\le 1, 0\le c\le 2, 0\le d\le 1\}$$ There are $(4+1)\times(1+1)\times (2+1)\times (1+1)=60$ positive divisors.

c) $a^2$ corresponds to $(8,2,4,2)$. To make a cube, each must be a multiple of $3$, i.e. $(9,3,6,3)$. Subtracting, we get $(1,1,2,1)$, i.e. $2^15^17^211^1=m$.

d) For $a$ to divide $b^2$, $u$ must be majorized by the $4$-vector of $b^2$. For example, if $b$ corresponds to $(5,1,2,2)$, then $b^2$ corresponds to $(10,2,4,4)$, which majorizes $u$, since it is larger in each component. I leave to you to count how many there are of these.

b) You seek $\phi(43120)$, the Euler totient. This is multiplicative, so you want $\phi(2^4)\phi(5)\phi(7^2)\phi(11)$.

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  • $\begingroup$ +1. For the OP, to compute the factorization, it might help to observe that $a = 43120 = 40320+2800 = 8!+4*7*100$. Then $a = (1)(2)(3)(2^2)(5)(2*3)(7)(2^3)+(2^2)(7)(2^2*5^2) = (2^4)(5)(7)(2^3*3^2+5) = (2^4)(5)(7)(77) = (2^4)(5)(7^2)(11)$. $\endgroup$ – Alex Wertheim Dec 10 '14 at 20:03
  • $\begingroup$ Sure. Or just trial division by small primes, there are many ways. I did it with wolfram alpha. $\endgroup$ – vadim123 Dec 10 '14 at 20:05
  • $\begingroup$ You guys are so quick to help! Thank you so much! $\endgroup$ – Murray Dec 10 '14 at 20:24

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