Extended Euclidean Algorithm: table form

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Question statement

Applying the Euclidean Algorithm to the integers $\var{a}$ and $\var{b}$ yields the equations

$\var{calcs[0][1]}=\var{calcs[0][0]}\times\var{calcs[0][2]}+\var{calcs[1][2]},$

$\var{calcs[1][1]}=\var{calcs[1][0]}\times\var{calcs[1][2]}+\var{calcs[2][2]},$

$\var{calcs[2][1]}=\var{calcs[2][0]}\times\var{calcs[2][2]}+\var{calcs[3][2]},$

$\var{calcs[3][1]}=\var{calcs[3][0]}\times\var{calcs[3][2]}+\var{calcs[4][2]},$

$\var{calcs[4][1]}=\var{calcs[4][0]}\times\var{calcs[4][2]}+\var{calcs[5][2]},$

$\var{calcs[5][1]}=\var{calcs[5][0]}\times\var{calcs[5][2]}+\var{calcs[6][2]},$

$\var{calcs[6][1]}=\var{calcs[6][0]}\times\var{calcs[6][2]}+\var{calcs[7][2]}.$

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			Name	Type	Generated Value

a	integer	2550
b	integer	1465
calcs	list	Nested 8×7 list
k	number	7
quotients	list	[ 1, 1, 2, 1, 5, 1, 10 ]
smatrix	matrix	matrix([1,0,1,-1,3,-4,23,-27])
tmatrix	matrix	matrix([0,1,-1,2,-5,7,-40,47])

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Generated value: `integer`

2550

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Parts

Part a) Matrix entry
Part b) Matrix entry
Part c) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Matrix entry

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Compute the values $s_0,s_1,\dotsc,s_7$ described in Algorithm 2.9 in the Number Theory Unit 1 notes.

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smatrix

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Advice

a) As per the notes, we determine the values of $s_i$ by using the formula $s_i=s_{i-2}-qs_{i-1}$ , with $s_0=1$ and $s_1=0$ .

b) As per the notes, we determine the values of $t_i$ by using the formula $t_i=t_{i-2}-qt_{i-1}$ , with $t_0=0$ and $t_1=1$ .

c) The required values are $s_7$ and $t_7$ , i.e. $\var{smatrix[0][k]}$ and $\var{tmatrix[0][k]}$ . We can check that these values are correct by observing that $\var{smatrix[0][k]}\cdot\var{a}+\var{tmatrix[0][k]}\cdot\var{b}=\var{gcd(a,b)}$ as required.

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Part	Test	Passed?
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Matrix entry
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This question is used in the following exams:

testtest by Maura Paterson in Maura's workspace.
NTC Assignment 1 Quiz by Maura Paterson in Maura's workspace.

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Taxonomy: mathcentre

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Taxonomy: Context

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Maura Paterson 1 year, 6 months ago

Maura Paterson 1 year, 7 months ago

Maura Paterson 1 year, 7 months ago

Error in variable testing condition

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Generated value: `integer`

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Extended Euclidean Algorithm: table form

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Maura Paterson 1 year, 6 months ago

Maura Paterson 1 year, 7 months ago

Maura Paterson 1 year, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Matrix entry

Advanced settings

Generated value: `integer`