John's copy of Solve an equation in modular arithmetic

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John Moss 5 years, 10 months ago

Created this as a copy of Solve an equation in modular arithmetic.

Name	Status	Author	Last Modified
Solve an equation in modular arithmetic	Doesn't work	Newcastle University Mathematics and Statistics	20/10/2023 11:40
John's copy of Solve an equation in modular arithmetic	draft	John Moss	18/06/2019 10:55

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

r1	integer	42
mess	string	A positive inverse of $\var{sa
ans2	integer	36
ans3	integer	42
csa	integer	-13
csb	integer	4
r	integer	42
sb	integer	49
sa	number	15

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(10..sb-1)

JME function reference

Description

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

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Generated value: `integer`

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Parts

Gap-fill
1. Gap Number entry
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Gap-fill

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

Solve the equation:

$\var{sa}x\;\equiv\;\var{r}\;\mod\;\var{sb}$

$x=\;\;$ Unnamed gap

Your value of $x$ should satisfy $0 \leq x \leq \var{sb-1}$

Advice

We use the extended Euclidean Algorithm to get:

{mdescextgcd(sa,sb,1,0,0,1)}

It follows then that we have $\var{csa}\times \var{sa}\;\equiv\;1\;\mod \var{sb}$

{mess}

We are asked to solve

$\var{sa}x\;\equiv\;\var{r}\;\mod\;\var{sb}$

Since

$\begin{eqnarray*} \var{ans2}\times\var{sa}\;&\equiv&\;1\;\mod\;\var{sb} \Rightarrow\\ \var{ans2}\times \var{r} \times \var{sa}\;&\equiv&\;\var{r}\;\mod\;\var{sb}\Rightarrow\\ \var{ans2*r}\times \var{sa}&\equiv&\;\var{r}\;\mod\;\var{sb} \end{eqnarray*}$

Hence

$x=\var{ans2*r}\; \equiv\;\var{ans3}\;\mod\;\var{sb}$

is the solution in the range $0\le x \le \var{sb-1}$ .

Give a worked solution to the whole question.

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Part	Test	Passed?
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Practice Cryptography by John Moss in John's workspace.

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John's copy of Solve an equation in modular arithmetic

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Taxonomy: mathcentre

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Checkpoint description

John Moss 5 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

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John's copy of Solve an equation in modular arithmetic

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

John Moss 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`