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GRE Math Formula eBook

Based on your suggestions, our third Magoosh GRE eBook is a compilation of must-know GRE math formulas for you to reference and memorize as you study! Though memorizing a math formula (or twelve) isn’t the only way to study for GRE Quant, knowing certain frequently-tested formulas by heart will help you improve your speed on test day. Once you learn these formulas, be sure to practice using them! A little timed practice can go a long way.

Download the Magoosh’s Complete Guide to Math Formulas by clicking the image below.

math formulas, math formula, gre math formulas

GRE Quantitative Reasoning & Useful Math Formulas

The GRE Quantitative Reasoning section tests concepts that you likely learned during sophomore and/or junior year of high school. These concepts DO NOT INCLUDE higher level math like trigonometry, calculus, or geometry proofs. They DO INCLUDE things you’ve probably long forgotten like properties of shapes, integer properties, exponent rules, and word problems.

The thing that takes these concepts to the GRE level is their complexity. Figuring out what each question is asking you to accomplish can be really tricky, whereas the math involved in solving them tends to be fairly straightforward. And you won’t get partial credit for showing your work, so picking the correct answer choice is very important!

Since it may take you longer than you’d like to read and understand what a GRE quant question is asking if yuo, it’s important that you won’t waste time solving the problem. Knowing which math formulas to use, and then using them quickly and correctly, can really help you do well on GRE quant. Time is of the essence!

GRE Math Formula Topics

Here is a list of the main concepts and formulas you’ll need to know for the GRE. Each of these concepts is addressed in the Magoosh’s Complete Guide to Math Formulas eBook. In this resource, you’ll also find recommended strategies for how to best use (or not use!) and remember these math formulas on the GRE.

  • Arithmetic and Number Properties
  • Types of Numbers
  • Order of Operations: PEMDAS
  • Commutative, Associative, and Distributive Properties
  • Prime Numbers
  • Factorization
  • Divisibility
  • Absolute Values
  • Fast Fractions
  • Percentages
  • Ratios
  • Powers and Roots
  • Exponents
  • Roots
  • Algebra
  • Simplifying Expressions
  • Factoring
  • Solving Equations
  • Geometry
  • Angles
  • Polygons
  • Triangles
  • Circles
  • Squares
  • Rectangles
  • Trapezoids
  • Quadrilaterals
  • 3-D Shapes
  • Coordinate Geometry
  • Lines
  • The Distance Formula
  • Quadratics
  • Word Problems
  • Distance, Rate, and Time
  • Work Rate
  • Sequences
  • Interest
  • Statistics
  • Counting
  • Probability

Don’t be intimidated! Some of these formulas are easy to remember, and some you’ll still know from your high school days. Focus your time on re-learning the formulas and concepts that are trickiest and newest to you.

Important GRE Math Formulas

Below are the main math formulas you’ll need for the GRE. I’ve just given a quick overview — a Cliff Notes version — since later on in the post you’ll find the full Math Formula eBook, which contains these formulas and many more. So if you want an in-depth take on this stuff, then head to the eBook at the bottom of the page. If not, look below for your math formula edification.

That said, I should offer a quick word on the temptation of thinking that knowing a formula is the key to automatically answering a question correctly. This is decidedly not the case on the GRE, on which you have to decipher, unscramble, unlock (pick your metaphor!) the question before you can answer it correctly. In other words, the formula is the final step to answering a math question right. (I expound on this in the intro of the eBook below.)

1. Area and circumference of a circle

Area: pir^2, where r is the radius of the circle
Circumference: pid, where d is the diameter of the circle

2. Volume and surface area of a cube

Surface area: 6s^2, where s is the length of a side of the cube
Volume: s^3, where s is the length of a side of the cube

3. Area of a triangle

(bh)/2,where b is the base and h is the height of the triangle

4. Exponent essentials

x^Ax^B = x^(A+B)
(x^A)^B = x^AB
(x^A)/(x^B) = x^(A-B)

5. Polynomials and FOIL

(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2
(a-b)(a+b) = a^2 - b^2

FOIL: First Outer Inner Last

6. Total degree measure of a polygon

Total degrees = 180(n-2), where n equals number of sides of polygon

Download the eBook as a PDF or read it here!

You can download the Math Formula eBook as a PDF here or read it below!

We recommend that you print it out, share it, and start a pile of Magoosh eBooks (General, Vocabulary), that you can study on the go. Enjoy! Be sure to let us know you like this resource. ๐Ÿ™‚

Editorโ€™s Note: This post was originally published in June 2012 and has been updated for freshness, accuracy, and comprehensiveness.

By the way, students who use Magoosh GRE improve their scores by an average of 8 points on the new scale (150 points on the old scale.) Click here to learn more.

80 Responses to GRE Math Formula eBook

  1. Mas August 17, 2016 at 8:46 pm #

    Thank you for your help. I plan to take the test in three weeks. Could you please put the links of the best post in all test sections. The posts you feel it is a must to read. I have heard that the scores of the test have changed in terms of the minimum and maximum scores. What is new, please?
    Again, Thank you very much.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert August 18, 2016 at 8:53 am #

      Hi Mas,

      The GRE changed in 2011 from a 200-800 scale to the 260-340 scale it uses now. Other than that, there have been no recent changes! ๐Ÿ™‚

      • Mas August 19, 2016 at 5:55 pm #

        Thank you for your reply. I got the following files from your website:

        and this

        however, these posts are from 2011 or 2012;
        Do you have updated versions of these links above?

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert August 20, 2016 at 2:23 pm #

          Actually, both of those resources are from 2012. But point taken– these eBooks are a little on the older side. We haven’t updated them just yet, as changes tot he GRE since 2012 have been very minimal. In fact, the GRE exam and the skills required for it have remained pretty much unchanged since the 2010-2011 redesign of the test.

          It’s also worth noting that ETS itself hasn’t done an updated official guide since 2012 either! Until we hear of updates from the official makers of the GRE, we’ll likely be leaving our ebooks as-is.

          For the most up-to-date GRE materials and info, though, you can watch this blog, and keep an eye on the official GRE website.

  2. Salma July 27, 2016 at 3:00 pm #

    Hi Margarette,

    In the new GRE do I have to take all the three section? For my admission verbal and quantitative score is enough. Can I skip the analytical section? Please let me know.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert August 2, 2016 at 1:26 pm #

      If your university really doesn’t need nay kind of AWA score, then you can “click through” the AWA section and not type in any answer. You can’t skip it entirely and not have an AWA prompt appear in your GRE testing session though. I’ve worked with students who have been able to skip AWA in this way and still be accepted. But you may want to double check with the admissions office of your target school to make sure this is OK.

  3. May Nguyen July 20, 2016 at 12:36 pm #

    This is absolutely helpful for my test prep. Thank you so much!

  4. Rushil Sharma July 6, 2016 at 5:54 am #

    How much full focused time is needed for preparing GRE? By focused preparations I mean just GRE all day .

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert July 13, 2016 at 12:48 pm #

      That really depends on where your GRE skills are at when you begin to study. I can tell you that it’s much easier to fully prep for the GRE in less than 3 months and even as little as one month, if you can study all day, at least 6 days a week. But again, your mileage may vary.

      • Rushil July 21, 2016 at 1:02 pm #

        Thanks for replying! ๐Ÿ™‚ I got 166 in math and 153 in verbal in ETS powerprep. My RC section doesn’t seem to improve ๐Ÿ™

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert July 22, 2016 at 7:16 am #

          Hi Rushil,

          Firstly, you’re very welcome ๐Ÿ˜€ And nice job on the Quant section of the PowerPrep exam! That’s an excellent score ๐Ÿ™‚

          In terms of improving on RC, we have a whole section on our blog focused on GRE RC: GRE Reading Comprehension. You’ll find links to posts on the different types of RC questions, strategies, and study tips. And here are a few articles that I highly recommend:

          * Introduction to Reading Comprehension (Strategies and Pacing)
          * How to Approach Questions
          * Pacing Strategies

          In addition to practicing the strategies discussed in these resources, reading is an excellent way to improve your RC skills and Verbal score! For some reading suggestions and how to take advantage of your outside reading, check out this post ๐Ÿ˜€

          Hope this helps! Happy studying ๐Ÿ™‚

          • Rushil August 10, 2016 at 7:54 am #

            Thanks for all the tips! I gave my gre and got 326 . 162 in verbal and 164 quant. My quant scores are not up to the mark , would it be advisable to give gre again? Again, I cannot thank you guys enough for my verbal score!

            • Magoosh Test Prep Expert
              Magoosh Test Prep Expert August 10, 2016 at 1:06 pm #

              Hi Rushil,

              If you think you can improve on that score and you need a higher score to get where you’re going, then we definitely would recommend taking it again. To be honest, though, a 164 is not a bad score! (I know that sometimes it doesn’t get you where you want to go depending on your program, though.) Good luck! ๐Ÿ™‚

  5. Kay June 17, 2016 at 10:47 pm #

    Awesome information thnq so much magoosh……

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 18, 2016 at 2:39 am #

      You’re very welcome, Kay! ๐Ÿ™‚

  6. Ragini June 13, 2016 at 4:05 am #

    hello sir,
    I’m studying in 2nd year engineering.Should I start preparing fr GRE from now?
    Please guide me how t start to study.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 16, 2016 at 9:49 am #

      Hi Ragini,

      Typically you don’t need to prepare that early for the GRE, but if it fits in your schedule and is something you want to do, there’s no reason why not! This early, your best plan of attack would be to get a lot of reading practice, learn vocabulary, and really work through any mathematical fundamentals that the GRE expects that you may not be comfortable with (such as combinations and permutations).

      I would not stress full-on GRE prep quite yet, but you can give yourself a huge leg up if you cover that basic, fundamental ground now. Then your only focus will be GRE style and trickier concepts. ๐Ÿ™‚

  7. Samy May 4, 2016 at 9:47 am #

    i am going to take GRE in Aug2016.I am practising verbal for nearly 3months.But i am not fulfilled with that.It makes me little uncomfortable now.Can anyone suggest pls…

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 9, 2016 at 1:40 am #

      Hi Samy,

      Firstly, to improve your verbal score, we at Magoosh cannot stress enough how important it is to read as MUCH as possible. We recommend that you try to read at least half an hour a day of GRE-level reading materials, such as our selections for our “GRE Article of the Monthโ€ series. This reading time is in addition to your formal GRE studying. Reading will help you improve your understanding of vocabulary in context as well as your comprehension. To take advantage of your reading time, make flashcards of the vocab words you don’t know is a good idea and summarize the main messages of what you’re reading in your own words every so often. ๐Ÿ™‚

      In addition to reading more, you’ll want to continue to practice and really analyze your mistakes! As yourself why you got a given practice problem wrong and think about what you can do in the future to avoid similar mistakes. Plus, I’d recommend learning about 10 new vocabulary words a day using our GRE Vocab Flashcards or GRE Vocabulary eBook.

      I hope these suggestions help ๐Ÿ™‚

  8. Rakshith April 13, 2016 at 7:49 am #

    I’m willing to take up GRE in August 2016. The post and comments are dated of 2012. Will the same ebooks help me in preparing now? Or any upgrades versions are available. Got confusion because of dates on posts.. Please reply

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert April 14, 2016 at 12:53 am #

      Hi Rakshith,

      Thanks for reaching out! ๐Ÿ™‚

      We regularly introduce new blog posts and update our materials. In the case of this eBook, for example, there haven’t been a lot of adjustments needed. The formulas for mathematical topics don’t change regularly, and the GRE itself has not updated the broad types of math content it has, so a resource like this one doesn’t need to be regularly remade. ๐Ÿ™‚

      We are continually engaged with updated resources, though, and our book reviews posts will reflect this, showing yearly (or sometimes even more frequently!) updates as new editions of books are released, for example.

      Don’t worry! We haven’t let things get stale. ๐Ÿ™‚

  9. enidan3 March 14, 2016 at 12:53 am #

    I’m not sure if this problem is being faced by anyone else, but in the PDF version of the Math GRE Formula eBook there appears to be some issue with certain formulae and examples not appearing clearly/ as clearly as the written text. In some cases it is almost impossible to decipher. Any assistance with this would be much appreciated.


    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert March 15, 2016 at 1:25 pm #

      Hi there ๐Ÿ™‚

      Thanks for writing in about this! I just downloaded a copy of the PDF version to see if I would encounter similar issues. While I agree that the formulas in the eBook are less clear than the accompanying text due to the equation editor used when creating the eBook, I was able to read all of the formulas without an issue. That being said, I’ve let the content creators here at Magoosh know about your experience so that we can improve the eBook in future editions. However, since we have many projects going on, releasing an updated eBook may take some time. So, in the meantime, please let us know if there is specific formulas or examples that are particular hard to decipher, and we’d be happy to help you out with those ๐Ÿ™‚

      Happy studying!

  10. Ganga Gowri January 7, 2016 at 5:44 am #

    Hi magoosh Team,

    Could you please explain the following property.

    Any counting number including negative numbers (e.g. -3, -1, 2, 7โ€ฆbut not 2.5)


    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert February 8, 2016 at 8:34 am #

      Hi Ganga ๐Ÿ™‚

      It looks like you’re asking about the definition of an integer. “Counting numbers” refer to all positive numbers with no fractional parts, i.e. the set of numbers {1, 2, 3, …}. The name comes from the fact that these are the numbers that we use to count things. Notice how counting numbers do not include zero or negative numbers. So, in our definition of an integer, we also have to include the negative of counting numbers and zero. As I mentioned, numbers with fractional parts are not integers, so 2.5 (2 1/2) is not an integer. The rest of the numbers listed are whole numbers and are either counting numbers or the negative of counting numbers and are therefore integers!

      I hope this clears up the definition ๐Ÿ™‚

  11. Omar faruq May 28, 2015 at 1:58 am #

    From my little knowledge i think it is more helpful for beginner of GRE exam .for math preparation.

  12. Asem May 6, 2015 at 7:34 am #

    Page 10.
    |X| is not equal to X, for X<0, but rather is equal to -X.

  13. Laurel August 11, 2014 at 1:14 pm #

    Thank you so much for this helpful workbook. I haven’t done math in years and this is a good reminder. My only comment is that only a few of the formulas have examples with them, and it would be helpful to have at least one example for each formula. Thanks!

  14. Ganesh July 5, 2014 at 1:39 am #

    Hi Magoosh team,

    I really appreciate your work with the ‘GRE Math Formula’ ebook. Just started going through it and I’m loving crisp and clear explanations.

    One doubt though on page 9 under title ‘Factorizations’ you have explained a method to find how many factors given no. has. e.g. for 720 it is given no. of factors as 30. but that is the no. of only positive factors right? So for the given question we should mention sum of positive as well as negative factors right? so factors of 720 has 60 factors (+ve + -ve)

    Let me know if my approach is right or am I giving too much thought to a simple logic.

    Thanks & Regards

  15. Anne July 3, 2014 at 9:44 am #

    I hav recently started using your ebook. Thank you so much for the book/ download
    I started on Factorization. I have a question on it.
    You said if I were to find how many factor a number has, I can find its prime factorization (2^a x 3^b x 5^c)
    where I have 5 choices (0,1,2,3,4) for a, 3 choices (0,1,2) for b and 2 choices (0,1) for c.

    The next page, to find the GCF, is to take prime factorization..
    56 = 2^3 x 7 <— where does the 7 come from if I were to use your given formula?
    70 = 2^1 x 5^1 x 7 <— again, where does it come from? I do not understand..

    I have been scratching my head for this other question. If I were to follow your rules, how do you find prime factor and GCF for 24 and 108? If I use your formula there's no way I can get 108.. because I only have 3 choices for b (0,1,2)

  16. eileen January 28, 2014 at 4:39 pm #

    All you get when you try to download the pdf is the first page. Where can I get the whole book?

  17. Rishi December 24, 2013 at 10:47 am #

    Thank you guys.It is really good book and helping me a lot.

  18. RAJ November 20, 2013 at 9:02 pm #

    thank alot for all the materials, videos ebooks and various other stuff u guys r providing for the GRE….. magoosh i doing a great job…….thank you mangoosh….!!!!!

    • Chris Lele
      Chris Lele November 21, 2013 at 2:27 pm #

      You’re welcome!

  19. Syed Osama September 23, 2013 at 6:58 pm #

    very helpful material U guyz shared .. Great job and THanks alot ๐Ÿ™‚

  20. Hithesh June 25, 2013 at 5:54 am #

    I can’t believe you are giving all of these ebooks for free…Great job

  21. Vijetha June 3, 2013 at 12:17 am #

    Thanks a lot…
    you are providing great deal of helpful information and study materials.

    • Chris Lele
      Chris Lele June 3, 2013 at 11:30 am #

      Great! I’m glad they are helpful!

  22. Anindya April 27, 2013 at 11:45 am #

    You guys rule, no second thought on that.

    Wish you could take classroom coaching in India, just a wish.

  23. Karan September 4, 2012 at 11:16 pm #

    culdn’t hav had askd for more ๐Ÿ˜› but hav to say an outstanding work from all of u ๐Ÿ™‚

    • Margarette Jung
      Margarette September 5, 2012 at 1:59 pm #

      Thanks, Karan! ๐Ÿ™‚

  24. abrish July 30, 2012 at 10:27 am #

    very helpful site,books and teachers here ๐Ÿ™‚

    • Margarette Jung
      Margarette July 31, 2012 at 1:38 pm #

      Thanks, Abrish! ๐Ÿ™‚

  25. abrish July 30, 2012 at 10:26 am #

    very helpful your site ,books and teachers here ๐Ÿ™‚

  26. abrish July 30, 2012 at 10:25 am #

    very helpful book and u also mam ๐Ÿ™‚

  27. praneeth July 13, 2012 at 9:17 am #

    Well, my math is always better than verbal but i’m really worried about one area related to “medians , quartiles standard deviations” ;their calculation – can you please please suggest a good link or anything else

  28. SriHarsha June 10, 2012 at 9:29 am #

    Hey Chris, Can you come up with a 2 month study plan.. I registered my test date on sep 5
    I am going to buy premium package.
    Thank You

    • Chris Lele
      Chris June 11, 2012 at 3:24 pm #

      Hi Srihasha,

      Actually, we have a 90 day study guide, which you can compress slightly to accomodate 2-months time frame. A good rule, as to not overburden yourself, is to do at least 80% of the recommended questions on the guide.

      Here’s the link :):

  29. VNMaths June 10, 2012 at 8:07 am #

    Hi Margarette,

    Thanks for the Maths formulas book, I found it so helpful for me!

    Uhm, I just concern if there is a typo of the Quadratic formula?!


    • Margarette Jung
      Margarette June 10, 2012 at 12:25 pm #

      It should be fixed now, thanks for catching that! Let us know if you find anything else! ๐Ÿ™‚

  30. Tahlia June 9, 2012 at 11:02 pm #

    seriously love you guys!! THANK YOU!!

    • Margarette Jung
      Margarette June 10, 2012 at 12:24 pm #

      You’re very welcome, I’m glad we could help! ๐Ÿ™‚

  31. Pemdas@BTG June 9, 2012 at 5:47 am #

    i just started reading the formulas you compiled together. Excellent work, congrats
    very useful, especially for those busy with verbal these days and lazy for math ๐Ÿ™‚

    • Margarette Jung
      Margarette June 12, 2012 at 10:12 am #

      Haha thank you! We’re happy to help ๐Ÿ™‚

  32. Sammy June 8, 2012 at 5:42 pm #

    This isn’t germane to the particular topic but do you still do the ‘Practice question(s) of the week’?
    I looked at some of your archived blog material and I found the questions to be challenging and educational.

    • Margarette Jung
      Margarette June 8, 2012 at 5:47 pm #

      Hi, Sammy

      Yes, we discontinued Practice Question of the Week for a variety of reasons, but they were basically just questions and explanations from the database of practice questions in the Magoosh product– if you liked those questions, I’d definitely recommend signing up for a Premium Magoosh account! ๐Ÿ™‚


      • Sammy June 8, 2012 at 5:50 pm #

        Oh okay, thanks! I found most of them to be hard yet feasible. Was this the purpose? Did you guys specifically pick out harder questions for your weekly practice problem?

        • Margarette Jung
          Margarette June 8, 2012 at 5:55 pm #

          The weekly questions were usually Medium to Hard. Generally, we avoided posting up particularly Easy ones because those seemed to provide less value to students. It’s more fun to have a tough problem and talk about various ways to tackle it as opposed to a simple practice question with a very straightforward answer :).

  33. Sammy June 8, 2012 at 7:32 am #

    You guys did a great job! Definitely will look over this more closely and keep a copy until test day.

    • Margarette Jung
      Margarette June 8, 2012 at 4:36 pm #

      Hi, Sammy

      We’re glad you like it! Let us know if there’s anything else we can help you with ๐Ÿ™‚


  34. Craig June 8, 2012 at 6:39 am #

    All ofnthe ebooks you provide are wonderful. I do have one suggestion. Can you make an ebook that includes important little tips and or strategies about particular topics? I know it sounds complicated, but from all the notes I have taken, I can explain more, again this is just a suggestion. Kind of like little things may not know that are vital to each property. Thanks so much.

    • Margarette Jung
      Margarette June 8, 2012 at 4:33 pm #

      Hi, Craig

      Do you mean for concepts (ex: Geometry) or for question types (ex: Quantitative Comparison)? The Magoosh lesson videos are great for that, and you can always send us any questions about anything you find unclear. Let me know if I’ve misunderstood your suggestions! ๐Ÿ™‚


      • Craig June 8, 2012 at 4:55 pm #

        Kind of like knowing for example in a qc, mc, all that apply, etc, how can algebra, geometry, nmber property questions be asked?

        • Margarette Jung
          Margarette June 8, 2012 at 5:14 pm #

          Ah, I understand. That would be pretty tough, since each question can have a blend of concepts, with many, many combinations thereof. The best thing to do would be to try a lot of practice questions until you get a better feel for how to tell, at a glance, which concepts you need to know in order to answer it correctly.

          I hope that makes sense! Let us know if you have any other questions or suggestions ๐Ÿ™‚

          • Craig June 8, 2012 at 5:23 pm #

            Yea I am writing down that qc questions don’t ask must be statements. Knowing whether certain strategies always work in different question types, etc.

            • Craig June 8, 2012 at 5:26 pm #

              U guys should make an ebook on the math question area itself. Like the types of questions and the strategies for each

              • Margarette Jung
                Margarette June 8, 2012 at 5:32 pm #

                Aha, that I can help with! Check out the Math section in the General eBook:

                • DHEERAJ March 24, 2013 at 5:29 am #

                  In the topic “Factorization” on page 10, paragraph 2, the GCF is typed instead of LCM. The GCF of 4 and 6 is 2 not 12, I think.

                  • Margarette Jung
                    Margarette March 25, 2013 at 4:13 pm #

                    Hi, Dheeraj

                    Yes, that’s a typo, thanks for catching it! It’s been fixed :).


  35. abhay June 8, 2012 at 6:21 am #

    your efforts are laudatory. Deserving a paeon for sure.

    • Margarette Jung
      Margarette June 8, 2012 at 4:29 pm #

      Thanks, Abhay! ๐Ÿ™‚ Chris would definitely approve of you actively using those GRE words! ๐Ÿ™‚

    • Steve June 9, 2012 at 1:21 am #

      paean (just keeping you on your toes)

  36. Dinesh June 8, 2012 at 1:25 am #

    I see 81 Practice questions left to be attempted on my dashboard though i completed all the questions one week back. Are they new? :). If it is new content, then it is awesome.. ๐Ÿ™‚ Am absolutely loving the content.

    • Margarette Jung
      Margarette June 8, 2012 at 4:28 pm #

      Hi, Dinesh

      Yep, we just added a new batch of Sentence Equivalence questions! ๐Ÿ˜€


Magoosh blog comment policy: To create the best experience for our readers, we will only approve comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! ๐Ÿ˜„ Due to the high volume of comments across all of our blogs, we cannot promise that all comments will receive responses from our instructors.

We highly encourage students to help each other out and respond to other students' comments if you can!

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