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• Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
Message 1 of 11 , Jan 8, 2002
Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Phil Carmody [mailto:fatphil@...]
Sent: 08 January 2002 11:09

On Tue, 08 January 2002, "paulmillscv" wrote:
> Prove that if gcd(a,b) = d, then gcd( |a-b|,min(a,b)) = d. where
> || is the modulus function, and min(a,b) is the min function.

That's a bit noisy - gcd is symmetric, so gcd(a,b)=gcd(b,a). Therefore you
can simply assume WLOG[*] a>=b, and your expression becomes, with no need
for abs or min,
gcd(a,b)=d -> gcd(a-b, b)=d

The proof is almost not deserving of the word, as
d|a & d|b -> d|(a-b),
d|(a-b) & d|b -> d|a

Phil
[* WLOG = Without Loss Of Generality]

Don't be fooled, CRC Press are _not_ the good guys.
They've taken Wolfram's money - _don't_ give them yours.
http://mathworld.wolfram.com/erics_commentary.html

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• ... Oh no it isn t! As zero divided by any integer d gives a quotient of zero and a remainder of zero gcd(0,d)=d Paul
Message 2 of 11 , Jan 8, 2002
> From: Jon Perry [mailto:perry@...]

> Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

Oh no it isn't!

As zero divided by any integer d gives a quotient of zero and a
remainder of zero gcd(0,d)=d

Paul
• An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does not equal 0. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
Message 3 of 11 , Jan 8, 2002
An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does
not equal 0.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Paul Leyland [mailto:pleyland@...]
Sent: 08 January 2002 18:11

> From: Jon Perry [mailto:perry@...]

> Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

Oh no it isn't!

As zero divided by any integer d gives a quotient of zero and a
remainder of zero gcd(0,d)=d

Paul

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• What I meant was gcd(0,p)=p (which is the fishy result) Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
Message 4 of 11 , Jan 8, 2002
What I meant was gcd(0,p)=p (which is the fishy result)

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Jon Perry [mailto:perry@...]
Sent: 08 January 2002 18:16

An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1), and 0/0 does
not equal 0.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Paul Leyland [mailto:pleyland@...]
Sent: 08 January 2002 18:11

> From: Jon Perry [mailto:perry@...]

> Also, if a=b=d, gcd(a,b)=d, but gcd(0,d)=0.

Oh no it isn't!

As zero divided by any integer d gives a quotient of zero and a
remainder of zero gcd(0,d)=d

Paul

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• ... You are quite correct: it is an arbitrary definition, but it s one which is convenient and almost universally accepted in the field of mathematics. Note
Message 5 of 11 , Jan 8, 2002
> An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1),
> and 0/0 does
> not equal 0.

You are quite correct: it is an arbitrary definition, but it's one which
is convenient and almost universally accepted in the field of
mathematics.

Note the primality of 0 and 1 has nothing to do with the definition of
gcd(0,d) and, indeed, neither is prime: again by a definition which is
both useful and (almost) ubiquitous.

0/0 is undefined. It's defined to be undefined 8-)

Paul
• I think 0/0 should be defined as 0. 0 multiplied by any non-zero number gives 0, and I don t see why multiplying it by infinity should be any different. If
Message 6 of 11 , Jan 8, 2002
I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
gives 0, and I don't see why multiplying it by infinity should be any
different.

If this is not so, then I could take my zero Euro's, and divide them into my
zero people, and end up extremely wealthy.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Paul Leyland [mailto:pleyland@...]
Sent: 08 January 2002 18:22

> An arbitary defn surely, like 0 or 1 are prime (gcd(0,1)=1),
> and 0/0 does
> not equal 0.

You are quite correct: it is an arbitrary definition, but it's one which
is convenient and almost universally accepted in the field of
mathematics.

Note the primality of 0 and 1 has nothing to do with the definition of
gcd(0,d) and, indeed, neither is prime: again by a definition which is
both useful and (almost) ubiquitous.

0/0 is undefined. It's defined to be undefined 8-)

Paul
• ? gcd(0,137) %1 = 137 integer g is a divisor of integer f if and only there is an integer g such that f=g*h g=137 is a divisor of f=137 because there is an
Message 7 of 11 , Jan 8, 2002
? gcd(0,137)
%1 = 137

integer g is a divisor of integer f
if and only there is an integer g such that f=g*h

g=137 is a divisor of
f=137 because there is an integer, namely
h=1, such that f=g*h

g=137 is a divisor of
f=0 because there is an integer, namely
h=0, such that f=g*h

g=137 is the largest integer that is a divisor of
both 0 and 137, since it the largest divisor of 137.

hence gcd(0,137)=137

Some folk might not like that,
but any alternative is sure to be far worse,
structurally.

Notice that no-where have we divided by zero.

d
• 0/0 shouldn t be defined as zero, consider the following: What is x/x as x-- 0? What is (x*x)/x as x-- 0? What is x/(x*x) as x-- 0? This clearly shows that
Message 8 of 11 , Jan 8, 2002
0/0 shouldn't be defined as zero, consider the following:
What is x/x as x-->0? What is (x*x)/x as x-->0? What is x/(x*x) as x-->0?
This clearly shows that 0/0 can take the values 0, 1 and infinity, and
indeed cna be made to tak any value you care to name. So "undefined" is
definitely the best thing for it.

Michael.

> I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
> gives 0, and I don't see why multiplying it by infinity should be any
> different.
>
> If this is not so, then I could take my zero Euro's, and divide them into
my
> zero people, and end up extremely wealthy.
>
> Jon Perry
• Reminds me of the riddle: What is apple multiplied by car? Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
Message 9 of 11 , Jan 9, 2002
Reminds me of the riddle:

What is apple multiplied by car?

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Michael Bell [mailto:mike.d.bell@...]
Sent: 08 January 2002 20:24
To: Primes List

0/0 shouldn't be defined as zero, consider the following:
What is x/x as x-->0? What is (x*x)/x as x-->0? What is x/(x*x) as x-->0?
This clearly shows that 0/0 can take the values 0, 1 and infinity, and
indeed cna be made to tak any value you care to name. So "undefined" is
definitely the best thing for it.

Michael.

> I think 0/0 should be defined as 0. 0 multiplied by any non-zero number
> gives 0, and I don't see why multiplying it by infinity should be any
> different.
>
> If this is not so, then I could take my zero Euro's, and divide them into
my
> zero people, and end up extremely wealthy.
>
> Jon Perry

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