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This was how I solved it:Force the equation into the form c = ax + by, where c = gcd(a,b):4 = a*16 + b*121 possible solution to this linear Diophantine equation:4 = (4)*16 + (-5)*12 ; [a=4, b=-5]Multiply both sides by W/4 to get back to the original equation:W = W*16 + -5*(W/4)*12, so one solution to the original equation is:x0 = W;y0 = -5*(W/4);The family of integer solutions always follows the form:X = x0 + (b/c)*nY = y0 - (a/c)*nfor all integers n.Therefore:X = W + 3*nY = -5*(W/4) - 4*nFinally, in the algorithm, start with n= -W/3 (round up to the nearest integer).Stop the loop once it moves outside the valid range of X>=0 and Y>=0;int X, Y;  int n = -1*((W + 2) / 3) - 1; unsigned char wasValid; unsigned char isValid = 0, stop = 0;   while( !stop ) {  n++;  wasValid = isValid;   X = W + 3*n;  Y = -5*(W/4) - 4*n;   isValid = (X >= 0 && Y>=0);  stop = (isValid==0 && wasValid==1); }   // Final Solution X = W + 3*(n-1); Y = -5*(W/4) - 4*(n-1); This can be further improved if need be, but for all practical purposes is satisfactory.The question may be useful in the context of SRAM array design. In certain cases, for purposes of maximizing area density, and minimizing the number of physical VLSI templates, memory sub-arrays are only available in 2 bit widths: width 12 bits and width 16 bits. Using the answer to the question, one can then then develop their SRAM compiler to support any bit width that is a multiple of 4 bits, except for 4, 8 and 20 (in which case it’s acceptable to round up to 12 or 24 bits).

Let’s start with a short story on how I learned about what quant finance is and what quants do. (Also let me make it clear that I have not in any way whatsoever mastered quant finance at all—have not gotten near that, so “How did Theodore Weld Smith learn quant finance” is a bit difficult to address as it isn’t a done-deal at all, rather a continuous process, but I’ll describe here how I’ve accomplished this task thus far).I was sitting in a frozen yogurt shop on Euclid talking with my dad about the professions people with math or physics degrees pursue outside of academia, and he mentioned that you could become a quant at a hedge fund—and he spoke of the famous quantitative hedge fund Renaissance Technologies and how their employee base consists of Princeton physics PhDs. This happened approximately three years ago.After hearing of this I looked into Renaissance Technologies and learned as much about them as I could on the internet, much of it here on Quora from people such as Vladimir Novakovski. Then I opened up a brokerage account and started doing some trading on my own, and was pretty successful.I continued to fully immerse myself in the subject, as I do with anything that I become interested in. My original interest was in algorithmic trading, which quantitative finance and the jobs of quants plays into. Quantitative finance is really just applied math—but it implies the programming implementation as well in 2019 (most quants know how to code too—and not just R or Excel).I started with looking through the internet for general information on the topic, here are the mediums that came up frequently:Quorar/algotrading (I’d recommend some caution on this subreddit now as its quality has declined severely, it used to be good but unfortunately now is filled with questions about how to use cryptocurrency APIs).Quantitative Finance Stack ExchangeNuclear PhynanceQuantopian I used extensively before I decided to invest in creating my own environment for developing and testing trading algorithms. I’d recommend it for any beginner looking to get into the field.QuantConnect I also used a bit for the same reasons.various news articles on algorithmic trading and HFT from places such as Forbes or Bloomberg.Then, I did something that is the largest distinction by far with any other of the subjects I have had intense interests in (physics and aerospace engineering as examples): I talked to people! I initiated conversations with tons of people on the internet. When you’re a fourteen year old asking questions about the significance of the efficient market hypothesis and how log normal returns are represented by geometric Brownian motion and how that plays into pricing derivatives… you tend to get people’s attention. Sometimes in a negative way, but overwhelmingly it has been positive. My success rate was (and still is) astoundingly high! Here were some of the places where I found most of these people:QuoraFacebook (often it was friends of someone who was a quant that I had already been interacting with; a lot of my relationships with people in the finance field can be traced back to 4 or so people).RedditLinkedIn (mostly Citadel, Jump Trading, and Jane Street people, I sent many of them connection requests and started talking with them).Quantitative Finance StackExchangeI then explored the topics that exist inside of quantitative finance, i.e., the subjects that apply to quant finance. Here are some of those topics and the sub-topics within them.Probability TheoryStochastic ProcessesStochastic calculus (lots of Ito’s Lemma[1])Markov Chains (yes, a kind of stochastic process)Brownian motion (same case as iii)Expectation value & confidence intervalsLinear AlgebraHessian MatrixHamiltonian Matrices (good stuff on this in the MCMC book).Stochastic Matrices (covered well in chapter six of the second book mentioned below).C++Concurrency & parallelismMemory ManagementTemplate meta-programmingdealing with large matrices and linear algebra computationsdealing with trading APIs (most prominently Interactive Brokers)The QuantLib C++ library for derivatives pricing.Pythonimplementing the various aforementioned math concepts programmaticallyvarious libraries within scipy and numpy, e.g., numpy.fft[2] , scipy.signal[3], etc.machine learning libraries in Python (mostly sklearn[4]), I plan on trying to learn how to use Tensorflow, though.Machine learning (covered a bit in the Python section, but here’s more)Recurrent neural networksLSTM models, (a sect of i)Sequence to sequence (commonly reffered to as Seq2Seq) learning with neural networksThere are probably more things that I’ll edit and include later, this is currently all in my mind.Here are some of the books I used along this process, starting with probability theory and statistics. I must thank my dad yet again for this as approximately 86% of the books mentioned (in all sections—not just probability) are his, many of which I “temporarily borrowed”. Also don’t think that just by virtue of these texts being mentioned that I understand everything from them, as this is not necessarily the case. I am merely putting them here as I have / am consulting them for my learning process of the quant field.Introduction to Probability and Statistics for Engineers and Scientists: Sheldon M. Ross: BooksIntroduction to Stochastic Models: Second Edition (Dover Books on Mathematics) : Roe Goodman, MathematicsStatistical Learning with Sparsity: The Lasso and Generalizations (Chapman & Hall/CRC Monographs on Statistics and Applied Probability) : Trevor Hastie, Robert Tibshirani, Martin WainwrightStatistical Decision Theory and Bayesian Analysis (Springer Series in Statistics): James O. BergerHandbook of Markov Chain Monte Carlo (Chapman & Hall/CRC Handbooks of Modern Statistical Methods): Steve Brooks, Andrew Gelman, Galin Jones, Xiao-Li MengProbability, Random Processes, and Estimation Theory for Engineers by Stark, Henry, Woods, John W.(February 23, 1994)Linear algebra.Numerical Linear Algebra: Lloyd N. Trefethen, David Bau IIIMatrix Computations (Johns Hopkins Studies in Mathematical Sciences)(3rd Edition): Gene H. Golub, Charles F. Van LoanApplied Matrix Algebra in the Statistical Sciences (Dover Books on Mathematics): Alexander BasilevskyC++.C++ Primer (5th Edition): Stanley B. Lippman, Josée Lajoie, Barbara E. MooThe C++ Programming Language (hardcover) (4th Edition): Bjarne StroustrupC++ Concurrency in Action: Practical Multithreading: Anthony Williams (all based on stuff from the C++11 standard, a newer version came out just four days ago that covers content from the C++14 and C++17 standards).I do not have any books on templates at the moment and various other important topics, I may purchase some soon but a great deal of my learning with those topics has been just all over the internet, a favorite spot of mine is cppreference.com.Math (other)Ordinary Differential Equations (The MIT Press): V.I. Arnold, Richard A. SilvermanA First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics): H. F. Weinberger:Applied Multivariate Analysis (Springer Texts in Statistics): Neil H. TimmProgramming (other)Real World OCamlI think the main reason I have been successful with this has actually nothing to do with programming or math skills at the moment: it’s about taking social risks. Creating these various internet accounts, interacting with adults, venturing into professional adult-dominated spaces, etc. you really stand out as someone my age and it’s a good thing. “and don't be afraid of taking social risks - they dont matter in the long run” to quote Alex K. Chen. It is one of the keys to success.After I did all this exploration (which I continue everyday), I started writing about it here on Quora and various other mediums (mostly the Quantitative Finance StackExchange and the r/highfreqtrading subreddit I made and moderate).Hopefully this is interesting / useful for some.ALSO: If anyone has any recommendation for further reading in the math (specifically stochastic processes section), I’d appreciate some books. Preferably unrelated to finance, actually.Footnotes[1] Itô's lemma - Wikipedia[2] Discrete Fourier Transform ([3] Signal processing ([4] scikit-learn: machine learning in Python

Note : Speaking and writing strategies mentioned below are pretty effective.Well it depends on whether your English is good or not and how much time you have till the test. By English I mean the Reading, Listening, Writing, and Speaking skills.My English is above average if all the 4 skills are considered.I scored 104 (L24,R29,W24,S27) by preparing for 7 days.According to me a working professional should have 6 hrs of study time daily.Work = 9 hrsSleep = 8 hrsOther = 1 hrStudy = 24-18 = 6 hrsTOEFL has four sectionsL- Listening : This section can be aced by a person whose English is average or above average by practicing 5-8 questions. You may end up spending 8 hours for practice.My StrategyI just practiced 4 questions, that's all. I didn't use any strategy. I felt comfortable answering this section. Yes my score is only 24 that's because i got an experimental question. You usually get an experimental question either in Reading or Listening.R- Reading : This section can be aced by a person whose English is average or above average by practicing 6-8 reading essays. The strategies you apply are the key. You may end up spending 12 hours for practice. The RC in TOEFL is lighter than in GMAT/GRE/CAT. So people who already prepared for these exams can relax.My StrategyMy strategy was to read first 2 paras. When i do so i will know about what the passage actually is about. So once i finish reading the 2 paras i directly go to questions. One nice thing about TOEFL Reading is along with the question the para that contains the answer also is shown. Mostly first 3 question's answers are in 1st 2 paras. From 4th question on-wards you read the question , read the respective para and then mark the answer.W-Writing :This section can be aced by a person whose English is average or above average by following the Notefull templates with in 5 days. 2 days(3 hrs daily) for getting familiar with writing templates. 3 days(3 hrs daily) for practice.My StrategyI followed the notefull writing templates only.S- Speaking :This section can be aced by a person whose English is average or above average by following the Notefull templates (see below) with in 5 days . 2 days( 3 hrs daily) for learning templates and 3 days( 3 hrs daily) for practice.My StrategyI followed the Notefull speaking templates only.Study time availableLets assume you have week's time till the test , so you will have6*5 = 30 hrs of study time during weekdays10*2 = 20 hrs of study time during weekendsTotal = 50 hrsPreparation timeListening : 8 hrsReading : 12 hrsSpeaking : 15 hrsWriting : 15 hrsTotal = 50 hrsHence in one week you can score more than 100. The above is a tested theory and it works. All the best !