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my pleasure. i too am a grad student in applied math and all we do is diffeq. my first year we had a whole sequence on greens functions and i really really loved it. Its funny how quickly it becomes impossible to find resources for these kinds of problems. I highly recommend this and this. I also think that latex is the best thing since sliced bread. are you in a regular math program or applied?

For proofs in general, I like D'Angelo and West's Mathematical Thinking. http://www.amazon.com/Mathematical-Thinking-Problem-Solving-Proofs-Edition/dp/0130144126

For discrete math, especially combinatorics, I loved Miklos Bona's A Walk Through Combinatorics. http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231/

For induction proofs, you check your base case, assume the induction hypothesis (true for k), and then check k+1.

You should be able to manipulate the k+1 term into something involving the k term, and that will then lead to the k+1 conclusion.

Example For all n >= 4, 2^(n) < n!

Base case: n = 4. 2^(4) = 16 < 24 = 4!

IH: Assume true for some k >= 4.

Then 2^(k+1) = 2*2^(k)

2*2^(k) < 2*k! (Induction Hypothesis used here)

2*k! < (k+1)k! (k > 3, so k+1 > 2)

(k+1)k! = (k+1)! (definition of factorial)

Here's a book you might enjoy. Well worth twelve bucks. Plenty of great material here for a paper.

http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347

The first one is a logic question. Specifically propositional logic, though if you are interested in logic you could also study basic predicate logic.

The second one could fall into a few overlapping categories, but I might put it into abstract algebra. I found a very good book that I used to self-prepare for a graduate course was this one by Dan Saracino. You may need to go one level more basic depending on your background, in which case i've been told this book is good.

This book refocused my life. It gave me the recognition of the value of my CS degree I did not have while I was doing my degree.

After I read GEB I read this:

https://www.amazon.com/gp/aw/d/0674324498/ref=mp_s_a_1_1?ie=UTF8&qid=1524603692&sr=8-1&pi=AC_SX236_SY340_QL65&keywords=frege+to+godel

I didn’t understand most of it and I didn’t follow most of the proofs, but reading the words of these men was quite a wild ride because I knew where the story would end, and reading the arguments between these brilliant people and seeing how each was so convinced of their view, and how wrong they were, and how some had grace and others lacked it... really fascinating thing to bear whiteness to.

Have you read Gödel, Escher, Bach? There are some great analogies between math, music, art, and other really neat stuff.

https://www.amazon.com/Gödel-Escher-Bach-Eternal-Golden/dp/0465026567

http://www.amazon.com/The-Art-Craft-Problem-Solving/dp/0471789011

This great book by Paul Zeitz contains chapter called "Algebraic manipulation revisited", I find it to be wonderful. You also probably won't find it in most "normal textbooks".

Did what? You said:

> x^(2) ≡ 4 (mod 8) when x is even is the problem

/u/picado gave a counterexample to that statement, showing that it's not true for any even x. :)

As for where to start on a problem like this, there's no universal answer. If nothing jumps out at you, try playing around with some small examples. What counts as "small" depends on the problem.

In this case, let's see what happens if we square the first few even numbers and look at their values mod 8. Maybe we'll notice something new that gives us an idea of what's "really happening." If we're lucky, we can take that idea and start turning it into a rigorous proof.

The purpose of doing this isn't to come up with the proof, but to make the shape of the problem less fuzzy.

n | n^2 | n^2 (mod 8)
----+-----+-------------
0 0 0
2 4 4
4 16 0
6 36 4
8 64 0
10 100 4

Anything stand out to you there? Any thoughts about where you could start with a proof? If the answer to either of those is "yes," you now know something about the problem you didn't know before! Hooray! Next time you can do this yourself instead of having someone like me suggest it. :)

Also, please don't think of this as the "use a table" trick or anything like that. This is a general problem-solving strategy. If wake up one day disoriented and alone on a desert island, the first thing you should do is get your bearings. If you don't know how to start tackling a problem, try to become more familiar with the problem in the simplest way you can.

I highly recommend Polya's How to Solve It.

Yeah, all these questions specifically say to use Cramer's rule. This is from Pre-Calc book by Dugapolski (Amazon LINK) (I think pretty common in US colleges). Probably the worst Math book Ive ever used. Doesnt explain things well, most of the time by just doing examples and not really listing steps of how to do things. You are supposed to look at the steps and figure out how they got there.

The worst offense is that it always uses super easy problems as examples. In this case for example... they use super easy linear equations to show Cramer's rule... not once they show how to handle x^2 or y^2. Yet in homework problems as you can see they ask you to solve them.

Youtube basically taught me everything so far. Class started with 30 people, we're down to 12 now (people dropping the class). There are 2 A's 2B's 4 C's and rest are F's ... because of this stupid book. (I am one of the A's)

Elementary Differential Equations and Boundary Value Problems by William Boyce

Don't bother getting that edition though. The latest editions are always overpriced and there's not much difference except for a few fixed typos. I used the 8th edition myself, and a quick google showed it was $5-$20 for it on ebay.

I would print out your course objectives page and a copy of your homework, take it to the department chair and ask him to deal with it. You'd never find those problems in an undergrad ODE book.

It's just called "algebra" by I.M. Gelfand and another dude.

http://www.amazon.com/Introduction-Mathematical-Statistics-Applications-Edition/dp/0321693949

This one

I found this book below. Do you think it'd be an overkill to study it?

Linear Geometry by Gutenberg&Weir

I'm not having any luck with your graph theory problem, but I can help with your paper problem:

http://www.amazon.com/Dry-Erase-Board-Melamine-Satin-Finished-Aluminum/dp/B002KO9JB0/

I own three whiteboards this size. I love them.

a, c and f can't be 0 because they are the first digit. As already stated, b and e can't be zero. If b or e were 0 then i would be zero.

Now, you have a two digit number, ab, multiplied by a 3 digit number, cde, these make a 4 digit number, fghi. So, a and c can only be 2, 3 or 4. In fact a and c can't be 3 and 4 together. One of them has to be 2. So a and c are 2 and 3 OR 2 and 4. This also means that f is 6, 7 8 or 9.

Let me just explain that if and and c were any bigger than 2, 3 or 4 then fghi would be a 5 digit number.

At this point it's getting late, I would try a little guessing and trial and error. I did google the problem and discover it's in this book http://www.amazon.com/Puzzles-Math-Logic-Dover-Recreational/dp/048622256X

Here's my advice. Buy the book. It's 1c plus shipping on Amazon. If your teacher has given you this problem, chances are he'll give you another from the book in a month's time.