Solve a pair of congruences

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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

pans	number	581
ansp1	integer	9
sb	integer	19
r1	integer	16
temp	number	399
csd	integer	1
s1	integer	7
csa	integer	9
csc	integer	-1
csb	integer	-8
a1	integer	11
s	integer	7
r	integer	16
ans	number	182
sc	number	20
ansp2	number	41
sa	number	17
invd	integer	-9
invb	integer	10
a2	number	14
sd	number	21

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a1*sd*invd+a2*sb*invb

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Description

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

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Generated value: `number`

581

← Depends on:

a1
a2
invb
invd
sb
sd

→ Used by:

Advice

Parts

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1. Gap Number entry
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1. Step Information only

Gap-fill

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Solve the equations:

$\begin{eqnarray*} \var{sa}x\;&\equiv&\;\var{r}\;\mod\;\var{sb}\\ \\ \var{sc}x\;&\equiv&\;\var{s}\;\mod\;\var{sd} \end{eqnarray*}$

$x=\;\;$ Unnamed gap

Your value of $x$ should satisfy $0 \leq x \lt \var{sb*sd}$

Advice

Using the method given in Steps:

First we find:

The inverse of $\var{sa}$ in $\mathbb{Z}_{\var{sb}}$ is $\var{ansp1}$ , the inverse of $\var{sc}$ in $\mathbb{Z}_{\var{sd}}$ is $\var{ansp2}$ .

(You can use the Extended Euclidean Algorithm to do this, and and your answers may differ from those above, but must be the same $\mod\;\var{sb}$ and $\mod\;\var{sd}$ respectively.)

Then multiplying both sides of the first equation by $\var{ansp1}$ and both sides of the second equation by $\var{ansp2}$ gives the pair of equations:

$\begin{eqnarray*} x\;&\equiv&\;\var{a1}\;\mod\;\var{sb}\\ \\ x\;&\equiv&\;\var{a2}\;\mod\;\var{sd} \end{eqnarray*}$

Now we use the formula given in Steps.

Let $n=\var{sb}\times \var{sd}=\var{sb*sd}$

Since $\var{sb}$ and $\var{sd}$ are coprime we find by using the Extended Euclidean Algorithm that:

1) The inverse of $\var{sb}\;\mod\;\var{sd}$ is $y_1=\var{invb}$

2) The inverse of $\var{sd}\;\mod\;\var{sb}$ is $y_2=\var{invd}$

(You may get different values, but they should be the same as those above $\mod\;\var{sd}$ and $\mod\;\var{sb}$ respectively.)

So $x=\simplify[std]{{a1}* {sd} * {invd}+{a2}*{sb}*{invb}}=\var{pans}=\var{ans} \;\mod\;\var{sb*sd}$

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Taxonomy: mathcentre

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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: `number`

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Solve a pair of congruences

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: number

Parts

Gap-fill

Generated value: `number`