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, 3 months ago

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Question statement

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			Name	Type	Generated Value

pans	number	-16
ansp1	integer	19
sb	integer	10
r1	integer	6
temp	number	90
csd	integer	-3
s1	integer	5
csa	integer	-1
csc	integer	4
csb	integer	1
a1	integer	4
s	integer	5
r	integer	6
ans	number	74
sc	number	7
ansp2	number	4
sa	number	9
invd	integer	-1
invb	integer	1
a2	number	2
sd	number	9

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Name

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Value

xxxxxxxxxx
 
a1*sd*invd+a2*sb*invb

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Generated value: `number`

-16

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Solve the equations:

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

$x=\;\;$ Unnamed gap

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

Advice

In order to solve the following two congruences we use the formula from Steps:

$\begin{eqnarray*} x\;&\equiv&\;\var{mod(a1,sb)}\;\mod\;\var{sb}\\ \\ x\;&\equiv&\;\var{mod(a2,sd)}\;\mod\;\var{sd} \end{eqnarray*}$
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 $x=\simplify[std]{{mod(a1,sb)}* {sd} * {invd}+{mod(a2,sd)}*{sb}*{invb}}=\var{pans}=\var{ans} \;\mod\;\var{sb*sd}$

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Taxonomy: mathcentre

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Newcastle University Mathematics and Statistics 9 years, 3 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, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`