Find n such that $\sigma(n)$ is the given number

Error

There was an error loading the page.

Contributors

Feedback

From users who are members of Content created by Newcastle University :

Christian Lawson-Perfect

said

Needs to be tested

5 years, 10 months ago

From users who are not members of Content created by Newcastle University :

Numbas Admin

said

Doesn't work

5 years, 10 months ago

History

○

Christian Lawson-Perfect 5 years, 10 months ago

Saved a checkpoint:

Part b seemed to be just some debugging info, so I've removed it

Restore this checkpoint

○

Christian Lawson-Perfect 5 years, 10 months ago

Gave some feedback: Needs to be tested

○

Numbas Admin commented 5 years, 10 months ago

Automatic testing has identified a problem with this question:
Part b got 0 credit

○

Numbas Admin 5 years, 10 months ago

Gave some feedback: Doesn't work

○

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

There is only one version of this question that you have access to.

Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

d12	list	List of 4 items
d124	list	List of 4 items
d18	list	List of 4 items
d217	list	List of 4 items
d24	list	List of 4 items
d28	list	List of 4 items
d39	list	List of 4 items
d403	list	List of 4 items
d42	list	List of 4 items
d78	list	List of 4 items
d91	list	List of 4 items
d93	list	List of 4 items

			Name	Type	Generated Value

biggest_prime	number	997
primes	list	List of 168 items
scenarios	list	Nested 12×2 list
scenario	list	[ 13, [ 9 ] ]
target	integer	13
answer	list	[ 9 ]
smallest_answer	integer	0
biggest_answer	number	9
target_factorisations	list	[ [ 13 ] ]
good_target_factorisations	list	[ [ 13 ] ]
target_one_more_than_a_prime	boolean	false
inverse_sigma_primes	list	[ [ [ 3, 2 ] ] ]
second_decomposition	boolean	false

			Name	Type	Generated Value

af1	integer	0
af2	integer	0
altf	string	There are no other factorizati
ans1	number	11
ans2	number	6
d	list	Nested 12×4 list
f1	integer	3
f2	integer	4
inv	string	, forgetting those with $2$ as
j	integer	0
m	integer	12
n1	list	[ 2, 1 ]
n2	list	[ 3, 1 ]
pri	string	Note that since $\var{m}-1$ is
v	number	6

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
[12,[3,4,1],[0,0],[[2,1],[3,1]]]

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`

[ 12, [ 3, 4, 1 ], [ 0, 0 ], [ [ 2, 1 ], [ 3, 1 ] ] ]

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. 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.

Find values of $n$ such that $\sigma(n)=\var{m}$ . There are at most 2.

If there are two values of $n$ enter in decreasing order.

If there is only one value for $n$ enter that value in the first answer field and enter 0 in the second field.

Largest value for $n = \;\;$ Gap 0.

Smallest value for $n=\;\;$ Gap 1. (enter 0 if there is no second value for $n$ )

Advice

First, recall that for a prime $p$ , $\sigma(p^\alpha) = \frac{p^{\alpha+1}-1}{p-1}$ , and that $\sigma$ is mulitplicative.

If $n$ is decomposed into powers of distinct primes, $n = p_1^{\beta_1}p_2^{\beta_2}\cdots p_s^{\beta_s}$ , then the sum of all divisors of $n$ is given by

$\begin{align} \sigma(n) &= \sigma(p_1^{\beta_1})\times \sigma(p_2^{\beta_2})\times\cdots \times\sigma(p_s^{\beta_s}) \\[0.5em] &= \frac{(p_1^{\beta_1+1}-1)}{p_1-1}\times \frac{(p_2^{\beta_2+1}-1)}{p_2-1} \times \cdots \times \frac{(p_m^{\beta_m+1}-1)}{p_m-1} \end{align}$

So for this example we have to examine how $\var{target}$ can be split into factors corresponding to such a prime decomposition.

$\var{target}$ can be factorised in the following ways:

$\var{latex(show_factorisations(target,target_factorisations))}$

Note that there is no $n$ such that $\sigma(n) = 2$ , so we can ignore the factorisations which involve $2$ .

Also note that if $\sigma(n)=m$ , and $m-1$ is a prime number, then we have $\sigma(m-1)=1+(m-1)=m$ so we have a solution immediately!

Since $\var{target}-1 = \var{target-1}$ is prime, $n = \var{target-1}$ is a solution.

In this case, $\var{m}-1$ is not prime, so $\var{m-1}$ is not a solution.

Now, we build up a table of values of $\sigma(q^\beta)$ for various primes $q$ and powers $\beta$ , stopping when $\sigma(q^{\beta})$ is bigger than any of the factors in our list.

$q$	$q^{\beta}$	$\sigma(q^{\beta})$	Value
$q=2$	$2^1$	$1+2$	$3$
	$2^2$	$1+2+4$	$7$
	$2^3$	$1+2+4+8$	$15$
	Too big!
$q=3$	$3^1$	$1+3$	$4$
	$3^2$	$1+3+9$	$13$
	Too big!
$q=5$	$5^1$	$1+5$	$6$
	$5^2$	$1+5+25$	$31$
	Too big!

We see from the table that

$\begin{align} \var{target} &= \var{good_target_factorisations[0][0]} \times \var{good_target_factorisations[0][1]} \\ &= \sigma(\var{inverse_sigma_primes[0][0]}^\var{inverse_sigma_primes[0][1]}) \times \sigma(\var{inverse_sigma_primes[1][0]}^\var{inverse_sigma_primes[1][1]}) \\ &= \sigma(\var{answer[0]}) \end{align}$

So $n=\var{v}$ is a solution.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only

This question is used in the following exams:

Loading...

Error

Find n such that $\sigma(n)$ is the given number

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 5 years, 10 months ago

Christian Lawson-Perfect 5 years, 10 months ago

Numbas Admin commented 5 years, 10 months ago

Numbas Admin 5 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

Loading...

Error

Find n such that σ(n)\sigma(n) is the given number

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 5 years, 10 months ago

Christian Lawson-Perfect 5 years, 10 months ago

Numbas Admin commented 5 years, 10 months ago

Numbas Admin 5 years, 10 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Find n such that $\sigma(n)$ is the given number

Generated value: `list`