Find $\mu(n)$, $\sigma(n)$, $\tau(n)$, $\phi(n)$

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

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			Name	Type	Generated Value

siga	list	[ 3, 1, 6, 1, 12 ]
eorodd	string	odd
phia1	list	List of 5 items
notzero	number	3
sig	number	216
siga1	list	[ 3, 1, 6, 1, 12 ]
ta	integer	8
phi	number	40
mess	string	and
a1	list	[ 1, 2 ]
a2	list	[ 1, 2 ]
a	list	[ 1, 2, 1, 2 ]
c	list	[ 1, 0, 1, 0, 1 ]
b	list	[ 1, 1, 2, 2 ]
mu1	number	3
d	list	[ 0, 0, 1, 1, 1 ]
phia	list	[ 1, 1, 4, 1, 10 ]
mu2	integer	1
n	number	110
mu	decimal	dec("-1")
p	list	[ 2, 3, 5, 7, 11 ]
s	integer	1
t	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
map(max(x,1),x,siga1)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ 3, 1, 6, 1, 12 ]

← Depends on:

siga1

→ Used by:

Advice

Parts

Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
1. Step Information only

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Let $n=\var{n}$.

Find:

$\mu(n) =\;\;$Gap 0.

$\tau(n)=\;\;$Gap 1.

$\sigma(n)=\;\;$Gap 2.

$\phi(n)=\;\;$Gap 3.

Advice

The factorization into a product of powers of prime factors is given by:

$\var{n} = \simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}}$

Finding $\mu(\var{n})$

There are an {eorodd} number $\var{notzero}$ of distinct prime factors, {mess} since the maximum power of any prime in this factorization is {d[4]} we see that

$\mu(\var{n})=\;\var{mu}$

Finding $\tau(\var{n})$

Recall that if $n=p_1^{\beta_1}\times p_2^{\beta_2}\times \cdots \times p_m^{\beta_m}$ is the prime factorization of $n$, then the number of divisors is given by:

$\tau(n)=(\beta_1+1)\times (\beta_2+1) \times\cdots \times (\beta_m+1)$

Hence in this case we have:

$\tau(\var{n})=\tau( \simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})=\simplify[!basic,unitFactor,zeroTerm]{({c[0]}+1)*({c[1]}+1)*({c[2]}+1)*({c[3]}+1)*({c[4]}+1)}=\var{ta}$

Finding $\sigma(\var{n})$

The sum over all divisors is given by :

\[\sigma(n)= \frac{(p_1^{\beta_1+1}-1)}{p_1-1}\times \frac{(p_2^{\beta_2+1}-1)}{p_2-1} \times \cdots \times \frac{(p_m^{\beta_m+1}-1)}{p_m-1}\]

In this case we have:

\[\begin{eqnarray*} \sigma(\var{n})&=&\sigma( \simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})\\ &=&\simplify[unitFactor]{{siga[0]}*{siga[1]}*{siga[2]}*{siga[3]}*{siga[4]}}\\ &=&\var{sig} \end{eqnarray*} \]

Finding $\phi(\var{n})$

This is the Euler totient function and $\phi(n)$ it counts up the number of integers less than $n$ which are coprime to $n$.

The formula we use for this is given by:

\[\phi(n) = (p_1^{\beta_1} - p_1^{\beta_1-1})\times (p_2^{\beta_2} - p_2^{\beta_2-1})\times\cdots \times (p_m^{\beta_m} - p_m^{\beta_m-1})\]

$\phi(\var{n})=\phi( \simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})=\simplify[unitFactor]{{phia[0]}*{phia[1]}*{phia[2]}*{phia[3]}*{phia[4]}}=\var{phi}$

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Find $\mu(n)$ , $\sigma(n)$ , $\tau(n)$ , $\phi(n)$

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Taxonomy: mathcentre

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Taxonomy: Context

Contributors

History

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `list`

Parts

Gap-fill

Finding $\mu(\var{n})$

Finding $\tau(\var{n})$

Finding $\sigma(\var{n})$

Finding $\phi(\var{n})$

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Find μ(n)\mu(n), σ(n)\sigma(n), τ(n)\tau(n), ϕ(n)\phi(n)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Finding $\mu(\var{n})$

Finding $\tau(\var{n})$

Finding $\sigma(\var{n})$

Finding $\phi(\var{n})$

Find $\mu(n)$ , $\sigma(n)$ , $\tau(n)$ , $\phi(n)$

Generated value: `list`