Question: Let $p$ and $q$ be distinct primes and let $n=pq$.  Prove that if $a=p+n\mathbb{Z}$ and $b=q+n\mathbb{Z}$, then $a^{q-1}+b^{p-1}=1.$  I'm not entirely convinced that the above is true.  I have applied Fermat's little theorem.  Since $p$ and $q$ are distinct primes - $a^{q-1}\equiv 1 \pmod q$ and $b^{p-1}\equiv 1 \pmod p.$  Where I am getting stuck is how to apply that second given: $a=p+n\mathbb{Z}$ and $b=q+n\mathbb{Z}.$ Added due to player3236: $a^{q-1}+b^{p-1}\equiv 1 \pmod q$ since $b^{q-1}\equiv 0\pmod q$.  Similarly for $a$ and $p$.  This shows that $a^{q-1}+b^{p-1}-1\equiv 0 \pmod {pq}$,or that $a^{q-1}+b^{p-1}-1$ is divisible by $pq$ since $p$ and $q$ are distinct primes.
I would like to prove the following equality: $$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}$$if $\gcd(l,k)=1$. What methods can I use? Thank you for your help.
What's the use of a decoupling capacitor near a reservoir capacitor? I've seen some circuits where a decoupling capacitor is used as well as a reservoir capacitor, like this (C4 and C5):    I've read about  and for me it looks as if they are meant to remove small fluctuations in the supply voltage. Then I thought - wasn't that the purpose of a  as well? Why wouldn't the reservoir capacitor be able to filter out the small fluctuations, if it is able to filter out the large fluctuations?  So I feel like I have a basic misunderstanding here. What is the purpose of a decoupling capacitor next to a reservoir capacitor, when we assume we place both equally near to the power consuming part? Or is the only advantage of the decoupling capacitor that it is smaller and can therefore be easily placed more near to the power consuming part? I am writing a bash script that needs to know which desktop environment (XFCE, Unity, KDE, LXDE, Mate, Cinnamon, GNOME2, GNOME3,... ) is running.  How can I obtain that information? Press-and-hold or Long-Press Gestures: Unintuitive? Often, I find people struggle with the usage of press-and-hold gestures on mobile/web platforms. The mere thought of having a user press on a defined object and hold for an arbitrary amount of time before the product reacts seems overtly complicated and unobtrusive. My questions are:   For what use cases do press-and-hold gestures best fit? Have there been published studies with regards to press-and-hold gesture intuition? Are there more viable alternatives to this system? How to call the method correctly in Preference.OnPreferenceChangeListener() I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? What are the use cases for different scoping constructs? Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $ Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... Slower download speed compared to Windows on Ubuntu 20.04 TRON: Game A.I algorithm? Where can I find the source code of Ubuntu? I would like to know where to find the source code of Ubuntu. I'd like to see how far it is "open source". What element saved Iron Man from his blood poisoning? I just watched Iron Man 1&amp;2 the other day. I'm new to the entire story. In IM2 we learn that his blood is being poisoned and that he has tried every combination of every existing element as a cure - unsuccessfully. Then he deducts &quot;the key&quot; from the Expo park's plans which saves him. What is that key? What element had he not tried? I realize that the movie story has to be simplified, compared to the original story, so there are probably story elements that I have completely missed because they're only in the original works. We also saw that in LOTR. Increase partition size on which Ubuntu is installed? I have Ubuntu installed on a drive. The problem is it is running low on disk space. This is installed on vmware. I expanded the virtual drive and booted into ubuntu. But when I opened gparted(sudo gparted), the move/resize option is unavailable. This is the partition Ubuntu is installed on, but I need to resize it. Any ideas? I am comfortable using command line How to make different materials for different objects? When I change one material, it changes it on another object. Why and how do I fix it? is there some link between the objects?    Here is my blender file of Iron Man's faceplate.  Here is my whole screen with the main portion of the face selected. On the right, you can see the material, and I can tell you that that is the same as the eyes part: ACPI PPC Probe failed. Starting version 219 NVIDIA implicit operator using interfaces I have a generic class that I'm trying to implement implicit type casting for.  While it mostly works, it won't work for interface casting.  Upon further investigation, I found that there is a compiler error: "User-defined conversion from interface" that applies.  While I understand that this should be enforced in some cases, what I'm trying to do does seem like a legitimate case.  Here's an example:  public class Foo&lt;T&gt; where T : IBar {     private readonly T instance;      public Foo(T instance)     {         this.instance = instance;     }     public T Instance     {         get { return instance; }     }     public static implicit operator Foo&lt;T&gt;(T instance)     {         return new Foo&lt;T&gt;(instance);     } }   Code to use it:  var concreteReferenceToBar = new ConcreteBar(); IBar intefaceReferenceToBar = concreteReferenceToBar; Foo&lt;ConcreteBar&gt; concreteFooFromConcreteBar = concreteReferenceToBar; Foo&lt;IBar&gt; fooFromConcreteBar = concreteReferenceToBar; Foo&lt;IBar&gt; fooFromInterfaceBar = intefaceReferenceToBar; // doesn't work   Does anyone know a workaround, or can anyone explain in a satisfactory way why I shuouldn't be able to cast interfaceReferenceToBar implicitly to Foo&lt;IBar&gt;, since in my case it is not being converted, but only contained within Foo?  EDIT: It looks like covariance might offer salvation.  Let's hope the C# 4.0 specification allows for implicit casting of interface types using covariance. AMPscript Error with XML I have a Partner sending some data to a Triggered Email/Data Extension which includes a column for XML data.   The trigger is erroring and not allowing the data to get to the Data Extension.   Below is what they are sending over. Is the formatting of the XML incorrect?  XML data that I've used in the past in Data Extensions starts with....   &lt;?xml version="1.0" encoding="UTF-8" standalone="yes"?&gt;   This version of XML starts with... {"XML":  Could that be it?  {"To": {"SubscriberKey": "test@acme.com", "ContactAttributes": {"SubscriberAttributes": **{"XML":** "&lt;root&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;dark chocolate nuts &amp; sea salt&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;15127&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/15127.html&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/nuts.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;crunchy peanut butter&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;41800&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/images/butter.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;11.37&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;9.55&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;pear cinnamon &amp; pecan&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;17227&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/apples.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;/root&gt;", "SHIPPING_LNAME": "Tester", "FIRST_NAME": "Test", "LAST_NAME": "Tester", "ORDER_TOTAL_SAVINGS": -5.97, "MERCHANT_ORDER_ID": "1810002352", "SHIPPING_ADDRESS_STATE": "NY", "ORDER_PLACE_DATE_LONG": "Fri 11 January 2019", "SHIPPING_ADDRESS_ZIP": "10004", "ORDER_SHIPPING": 0.00, "CARD_TYPE": "Visa", "ORDER_SUBTOTAL": 47.32, "ORDER_PLACE_DATE_SHORT": "01/11/2019", "SHIPPING_FNAME": "Test", "SHIPPING_ADDRESS_TOWN": "new york", "ORDER_TOTAL": 47.32, "CARD_EXPIRE": "12/2027", "CARD_HOLDER": "Test Tester", "SHIPPING_ADDRESS_1": "75 broad st", "SHIPPING_ADDRESS_2": "fl 23"}}, "Address": "acme@test.com"}}   AMPSCRIPT IN EMAIL Calling the XML....  %%[   var @LOOPINGAPPENDIX  set @LOOPINGAPPENDIX = AttributeValue("LOOPINGAPPENDIX")  set @productRows = BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM", 0)  ]%%   %%[   for @j = 1 to RowCount(@productRows) do       set @productXML = Concat("", Field(Row(@productRows, @j), "XML"), "")      set @ITEMPRODUCTNAME = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_NAME", 0), 1), "Value"))      set @ITEMPRODUCTSKU = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_SKU", 0), 1), "Value"))      set @ITEMPRODUCTURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_URL", 0), 1), "Value"))      set @ITEMPRODUCTIMGURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_IMG_URL", 0), 1), "Value"))      set @ITEMQTY = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_QTY", 0), 1), "Value"))      set @ITEMSUBSCRIPTIONPRICE = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_SUBSCRIPTION_PRICE", 0), 1), "Value"))      set @ITEMTOTAL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_TOTAL", 0), 1), "Value"))      set @rowColor = Iif(Mod(@j,2) == 0, "#efefef", "#ffffff")   ]%%   %%[endif]%%   %%[next @j]%% I know you can export the asset as a PSD and open it in Photoshop. Is there a more seamless way?  Update:  I'm an idiot.  Someone else created the logo, and the shadow was it's own layer on multiply and not a normal drop shadow effect. Replace arbitrary characters in the middle of an IP address string with sed I need to find and replace either one or two numerical characters in strings in a file.  The strings are IP addresses of the form:   10.xx.y.z    Where xx can be one or two characters.    I want to replace the xx with the single character 0, so I have  10.0.y.z    preserving the values of y and z.   The string may appear multiple times in the file.  What is the sed invocation to do this? How can you accelerate without moving?