Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$

Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$

If $a$ and $b$ are two coprime integers then prove that $\gcd((a+b)^m,
(a-b)^m) \leq 2^m$
My attempt:
I took two cases first one in which a and b are both odd then very clearly
the gcd will be 2^m but I'm stuck in the second case in which one of the
integer is even and the other one is odd I am reaching no where I tried
everything even binomial expansion but I'm stuck.

What should be teach about the reciprocity law for high school for gifted student

What should be teach about the reciprocity law for high school for gifted
student

This autumn I have to teach a mini course for a small group of high school
student (mathematical gifted class) on Quadratic Reciprocity Law (3
lectures in ten hours). This is the requirement of the school for me to
become an assistant lecturer. I have studied thís law in the second year
in university, but that was only a glimpse: the origin, the proof... and
my approach at that time was quite modern. I know that this law is very
beautiful and important in mathematics, but I did know much about its role
and applications.
So,
What are the books/lecture notes/papers should I read for preparing my talk?
What should be teach about this topic for high school student? (May be
like the reciprocity law with a view toward modern number
theory/mathematics?)

Assistance diagnosing SenderID PermError in Exchange 2010

Assistance diagnosing SenderID PermError in Exchange 2010

We're having issues with a customer's domain. They're wanting to do a
mail-out with a service called Act-On, and so far all the tests seem to be
getting flagged as spam.
The customer has tried sending to:
Themselves (Office 365): Goes to junk folder.
Gmail: Goes to junk folder.
Our Exchange: Gets quarantined.
So it seems clear there's an issue, and I believe it is SenderID, as in
our Quarantine mailbox, the NDR showed:
Received-SPF: PermError (exchange.ourdomain.com: domain of
person@customerdomain.com used an invalid SPF mechanism)
My issue is that I need assistance trying to figure out why it's giving
this error. The only tool that seems to be confirming the issue is
Exchange's own Test-SenderID cmdlet. Every other tool shows no issue.
According to Microsoft, and the OpenSPF docs, PermError should be some
kind of syntax or formatting issue. But I can't spot one, and none of the
tools I've used have hinted to one.
I've used the following SPF record, and also explicitly specified a
SenderID record in case this issue is at play.
;; QUESTION SECTION:
;customerdomain.com. IN TXT
;; ANSWER SECTION:
customerdomain.com. 2335 IN TXT "spf2.0/pra
include:spf.protection.outlook.com include:_spf.act-on.net -all"
customerdomain.com. 2335 IN TXT "MS=msxxxxxxxx"
customerdomain.com. 2335 IN TXT "v=spf1
include:spf.protection.outlook.com include:_spf.act-on.net -all"
What I've Tried
Checked for the SPF vs. SenderID issue.
Tested using this SPF Syntax checker:
http://www.kitterman.com/spf/validate.html - Passes
And this one: http://mxtoolbox.com/spf.aspx - Passes
Used the Microsoft SenderID Wizard to compare my SPF record, and then
generate the SenderID record - Seem to match, though interestingly, the
wizard never seems to detect the existing records.
As per this post I've checked both the Office 365 and Act-On SPF records I
am including, and they both seem valid.
And this:
http://www.port25.com/support/authentication-center/email-verification/ -
Passes
Below are details from the Port25 report - I asked for a copy of the
mail-out to be sent via Act-On as it would normally, so the email is
actually coming from Act-On (@b2b-mail.net):
==========================================================
Summary of Results
==========================================================
SPF check: pass
DomainKeys check: neutral
DKIM check: pass
Sender-ID check: pass
SpamAssassin check: ham
==========================================================
Details:
==========================================================
HELO hostname: mx139.b2b-mail.net
Source IP: 209.162.194.139
mail-from: delivery@b2b-mail.net
----------------------------------------------------------
SPF check details:
----------------------------------------------------------
Result: pass
ID(s) verified: smtp.mailfrom=delivery@b2b-mail.net
DNS record(s):
b2b-mail.net. SPF (no records)
b2b-mail.net. 3600 IN TXT "v=spf1 ip4:69.30.4.0/27 ip4:69.30.45.96/27
ip4:207.189.98.224/27 ip4:207.189.124.224/27 ip4:207.189.125.224/27
ip4:209.162.194.0/24 ~all"
----------------------------------------------------------
DomainKeys check details:
----------------------------------------------------------
Result: neutral (message not signed)
ID(s) verified: header.From=person@customerdomain.com
DNS record(s):

Update repositories on Ubuntu 8.04

Update repositories on Ubuntu 8.04

I just finished of installing ubuntu 8.04 on a low-resources computer.
When I try to update it says an repository error.
The support of 8.04 has finished, I suspect that the repositories too.
There are another repository for this version??

Get data for current month using Linq - VB

Get data for current month using Linq - VB

I have the following code which is listing the top 3 'names' in descending
order with a count of how many records that 'name' appears in in the
table. This is working ok.
Dim top3 = From s In dc.Suggestions
Group By Name = s.name
Into t3 = Group, Count()
Order By Count Descending
Take 3
I want to amend this so that it will only get a count of the number of
records each 'name' appears in for the current month. I can use
s.dateRaised to get the date of the record but i'm not entirely sure how
to write the code to get only the current month's records?

how to change attribute values using xsl

how to change attribute values using xsl

im still learning xsl and i'm trying to change the value im getting from
the xsl to another value. (im using xsl to convert my xml to another xml)
xml
<?xml version="1.0" encoding="UTF-8"?>
<Person>
<Info Name="Jen" Age="20" Class="C" />
</Person>
xsl
<?xml version="1.0" encoding="UTF-8"?>
<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
version="2.0">
<xsl:template match="/">
<Person>
<lookuptable>
<name first="Jen" ori="Jenny" />
<name first="Sam" ori="Sammy" />
</lookuptable>
<xsl:for-each select="Person">
<Info Name="{Info/@Name}" Age="{Info/@Age}" Class="{Info/@Class}" />
</xsl:for-each>
</Person>
</xsl:template>
</xsl:stylesheet>
i created a lookuptable. {Info/@Name} gives me "Jen" in the converted
xml(using xsl) i want to change it to "Jenny" using table but i dont know
how to do it. thanks in advance..

Allow user post products

Allow user post products

I need to know if I can in prestashop or another eccommerce framework
allow the users to publish his own products.
I ask this because i need that feature to my site.
Thanks in advance !