Found this after solving the 2020 exam for fun. This is the same solution as the one I came up with. But finding the actual solution by Putnam authors made me understand why we wouldn't get full points even for this solution. Instead of "non-pure-mathematical" 1...10...0 terms and a lot of explanation of what you mean by "alternating pattern" of mod(101), they use (10^a-10^b)/9 formula for N (where a<=2020), or 10^b*(10^(a-b)-1)/9 to check for divisibility. So we get the options of 10, -1, -10 or 1 for mod(101) with a-b being 1, 2, 3, or 0 mod(4), and NOW we get to the (a-b) being divisible by 4. So, b>1, a<=2020, a>b, and a-b is divisible by 4. They suggest that instead of filling with ones, we fill with zeroes and for every b (the number of zeroes) we have (2020-b)/4 possible values of a. and with 4(504+503+...+1)-504 (as b>1) we get the final answer. Coming up with an explanation THIS much math based is what they probably expect us to do, so having a simple explanation in this case probably won't work for full points, sadly:(
I don't understand the step at 22:23-22:39. If i rationalize i dont get what you said and if i split it into three fractions I dont get what you said so how did you get to the right hand side of the equality from the left hand side?
I'm not sure which equation you plugging things into, but it seems like you might be plugging ω = b into the first and then using the second equation: a ⊕ ω = a + ω and a ⊕ ω = a, and hence a + ω = a implying ω = 0. Is that right? If so, then the problem is that the first equation (a ⊕ b = a + b) is only defined for a, b ∈ ℝ and ω ∉ ℝ.
Your observation that we're trying to "get rid of redundancy" actually provides ample motivation for students with an applications bent. I'm an advocate of pure math, yet even I was appreciated the insight. New subscriber here. Thank you.
I know this video is 5 years old but why does the dimension being 2 let us know it's a basis? I know the minimal polynomial is x^2 - 7, but why does that allow us to say its a basis?
A general fact about vector spaces is it the length of any linearly independent list is always bounded from above by the length of any basis, and that the length of any spanning list is always at least as long as the length of any basis. Since (1,√7) spans Q(√7), it follows from the above fact that the dimension is no larger than 2. For the dimension to be less than two, the above fact implies that (1, √7) would have to be linearly dependent. However a list of length two consisting of nonzero vectors is only linearly dependent if the vectors are multiples of each other. In this case that would mean that √7 is a Q-multiple of 1, i.e., that √7 is a rational number-which it is not. Thus, the dimension must be exactly 2 and, since (1,√7) is linearly independent (and also spans Q(√7)), we conclude that (1, √7) is a basis.
This is a really nice example and a good practice question. Do you mind sharing the source of this problem and where to get this type of practice problems for galois theory?
@@AdamGlesser i am preparing to take a measure theory class this september. What concepts from real analysis are most important do you think? I was thinking integration, continuity and sequences of functions and convergence type things are probably most important. Differentiation, FTC, taylor series etc. isn’t that necessary for measure theory right?
@@ClumpypooCP I think you are on the tight track. Integration (the conceptual idea, not as much the formulas, e.g., integration by parts) is certainly the big motivation for measure theory. In addition to your list, I would also review some elementary point-set topology as well as basic notations such as inf, sup, lim inf, and lim sup.
@@AdamGlesser thank you! I am also taking a course in module theory. What is most important from abstract algebra should I review? I have not actually learned some important things from group theory like group actions, sylow theorem, structure theorem for abelian groups etc. is it most important to study those to prepare for module theory? Or should I mainly focus on reviewing concepts from ring theory?
@@ClumpypooCP The structure theorem for finite abelian groups is useful to study since you will likely generalize it to finitely-generated modules over a PID. Maybe the most important thing is to tighten up your linear algebra skills. Make sure you've seen things like Jordan Canonical Form and deeply understand the connection between matrices and linear maps. Being proficient with quotients (another thing you can study for vector spaces) could also be really helpful.
Since the length of C_k is (2/3)^k and since C is the intersection of the C_k (which are nested), we have that the length of C is the limit of the lengths of the C_k, i.e., the limit of (2/3)^k, which is 0.
Yes, but you may not like the answer. Because (1 2 3 4) is already a cycle with no repetitions, we consider it a product of disjoint cycles. I suspect you really want to know if we can write (1 2 3 4) as a product of two (or more) disjoint cycles. The answer to that is no. This is because, up to reordering the disjoint cycles, we can write any permutation as a product of disjoint cycles in only one way. Since (1 2 3 4) is already one such way, there are no others.
