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How to Edit and fill out Math 227 Online
Read the following instructions to use CocoDoc to start editing and writing your Math 227:
- At first, direct to the “Get Form” button and click on it.
- Wait until Math 227 is ready.
- Customize your document by using the toolbar on the top.
- Download your completed form and share it as you needed.
An Easy-to-Use Editing Tool for Modifying Math 227 on Your Way
How to Edit Your PDF Math 227 Online
Editing your form online is quite effortless. No need to install any software through your computer or phone to use this feature. CocoDoc offers an easy tool to edit your document directly through any web browser you use. The entire interface is well-organized.
Follow the step-by-step guide below to eidt your PDF files online:
- Search CocoDoc official website on your computer where you have your file.
- Seek the ‘Edit PDF Online’ option and click on it.
- Then you will browse this page. Just drag and drop the form, or attach the file through the ‘Choose File’ option.
- Once the document is uploaded, you can edit it using the toolbar as you needed.
- When the modification is finished, press the ‘Download’ button to save the file.
How to Edit Math 227 on Windows
Windows is the most widely-used operating system. However, Windows does not contain any default application that can directly edit form. In this case, you can install CocoDoc's desktop software for Windows, which can help you to work on documents effectively.
All you have to do is follow the instructions below:
- Download CocoDoc software from your Windows Store.
- Open the software and then select your PDF document.
- You can also upload the PDF file from URL.
- After that, edit the document as you needed by using the a wide range of tools on the top.
- Once done, you can now save the completed file to your cloud storage. You can also check more details about how can you edit a PDF.
How to Edit Math 227 on Mac
macOS comes with a default feature - Preview, to open PDF files. Although Mac users can view PDF files and even mark text on it, it does not support editing. Using CocoDoc, you can edit your document on Mac without hassle.
Follow the effortless instructions below to start editing:
- To begin with, install CocoDoc desktop app on your Mac computer.
- Then, select your PDF file through the app.
- You can select the form from any cloud storage, such as Dropbox, Google Drive, or OneDrive.
- Edit, fill and sign your file by utilizing this CocoDoc tool.
- Lastly, download the form to save it on your device.
How to Edit PDF Math 227 through G Suite
G Suite is a widely-used Google's suite of intelligent apps, which is designed to make your work more efficiently and increase collaboration across departments. Integrating CocoDoc's PDF file editor with G Suite can help to accomplish work easily.
Here are the instructions to do it:
- Open Google WorkPlace Marketplace on your laptop.
- Search for CocoDoc PDF Editor and download the add-on.
- Select the form that you want to edit and find CocoDoc PDF Editor by choosing "Open with" in Drive.
- Edit and sign your file using the toolbar.
- Save the completed PDF file on your laptop.
PDF Editor FAQ
What is the remainder of [math]\sum_{x=0}^{113}(226-2x)!(2x)![/math] when divided by 227?
First things first: [math]227[/math][math][/math] is a prime number, and so, by Wilson’s theorem, [math]226! \equiv -1\pmod{227}[/math].Now,[math](2x)! = 1\cdot 2\cdot 3\ldots (2x) \equiv(-1)^{2x}(227–1)(227–2)(227–3)\ldots(227–2x) = \frac{226!}{(226–2x)!}\pmod{227}[/math].Finally,[math]\sum_{x=0}^{113}(226–2x)!(2x)!\equiv \sum_{x=0}^{113}(226–2x)!\frac{226!}{(226–2x)!} = 113\cdot 226!\equiv114\pmod{227}[/math],and the required remainder is [math]114[/math].
When 3^45 is divided by 4^5, what's the remainder?
To make the numbers more manageable,[math]{3^{45}} \equiv {\left( {{3^9}} \right)^5}\bmod 1024[/math]But [math]3^9=19683[/math] and [math]19683 \equiv 227\bmod 1024[/math].So [math]{3^{45}} \equiv {227^5}\bmod 1024[/math][math]{227^5} \equiv 851\bmod 1024[/math]So the remainder is 851.
What are the ways to express 1000 as the addition of two prime numbers? How do I solve it?
If [math]p[/math] and [math]q[/math] are primes such that [math]p+q = 1000[/math] then one of [math]p[/math] and [math]q[/math] must be less than or equal to [math]500[/math]. Using a simple prime sieve, you find that there are only [math]95[/math] primes less than or equal to [math]500[/math]. For each of these [math]95[/math] primes [math]p_i[/math], test if [math]1000-p_i[/math] is also prime. This gives you the following [math]28[/math] ways to express [math]1000[/math] as the sum of two primes:[math]3+997[/math][math]17+983[/math][math]23+977[/math][math]29+971[/math][math]47+953[/math][math]53+947[/math][math]59+941[/math][math]71+929[/math][math]89+911[/math][math]113+887[/math][math]137+863[/math][math]173+827[/math][math]179+821[/math][math]191+809[/math][math]227+773[/math][math]239+761[/math][math]257+743[/math][math]281+719[/math][math]317+683[/math][math]347+653[/math][math]353+647[/math][math]359+641[/math][math]383+617[/math][math]401+599[/math][math]431+569[/math][math]443+557[/math][math]479+521[/math][math]491+509[/math]If you count [math]3+997[/math] and [math]997+3[/math] as two different ways, then we have [math]56[/math] ways to express [math]1000[/math] as the sum of two primes. Given that there are several solutions, there isn't likely to be an elegant way of finding all of them.
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