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This week was full Number Theory and algebra biased!! ðŸ˜„ Do try all the problems first!! And if you guys get any nice solutions, do post in the comments section! Here are the walkthroughs of this week's top 5 Number Theory problems! 5th position (1999 JBMO P2): For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. Walkthrough: It doesn't require a walkthrough, I wrote this here, cause it's a cute problem for the person who has just started Contest math :P a. What is $A_0$? b. Find out $A_1$. c. Show that $\boxed{7}$ is the required answer! 4th position (APMO, Evan Chen's orders modulo a prime handout): Let $a,b,c$ be distinct integers. Given that $a | bc + b + c, b | ca + c + a$ and $c | ab + a + b$, prove that at least one of $a, b, c$ is not prime. Walkthrough: Fully thanks to MSE ! (Also one should try MSE, it has helped me a lot, ofc it's more tilted to Coll

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In this blog post, we will talk about Combinatorial games, how to quantify positions, and analyse games. Most of this is based on the beginning chapters of the comprehensive book Winning Ways for your Mathematical Plays by Berlekamp, Conway and Guy. What is a Game? You might be familiar with multiple 'games', all with greatly varied meanings. For example, a game could refer to a wild animal hunted for sport (This post isn't about those!). We will talk about games that are defined by the following (restrictive) rules: There are just 2 players, often named Left and Right. There are several well-defined positions and a starting position. There are clearly defined rules specifying the moves each player can make from a position, and the resulting positions are called options. Left and Right move alternately in gameplay. In the normal play convention, the player unable to move loses. In misere play, the last player to move loses. The rules ensure that the game ends and one p

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Hi everyone! In my last blog, I spoke about a couple of very interesting algorithmic techniques and how they can be used to solve a variety of very interesting problems. This blog however, is going to be completely different. When we’re young we begin by learning the steps to add – we’re given the rules and we must learn to repeat them – no questions asked. Why does carry forward work the way it does? Well, no questions asked. To some extent, it is important to know how exactly to do a certain set of things while first learning maths. We may not be able to get anywhere in a subject if we’re unable to learn a few basic rules and know how to use them. However, after a certain point it is important to bring in the spirit of mathematical thinking within each student too – something missing in almost every form of school math education. Mathematical miseducation is so common, we wouldn’t even see it. We practically expect a math class to look like repetition and memorisation of disjointed

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The word 'Mobius' would instantly remind the readers of the Mobius strip, the famous object used to attract fellow students to the wonders of topology. But the discoverer of this object, August Ferdinand Mobius was also responsible for the creation of another amazing thing, the theory of Mobius inversions. It is a formula in number theory that answers the following question: Given two functions $f$ and $g$ on the set of naturals, if it is known that $$f(n)=\sum_{d\vert n}g(d),$$ how do you express $g$ in terms of $f$? Actually, the Mobius inversion formula that is applied in number theory is a special case of a more general theorem in the branch of Order theory. Without any doubt, we explore the general theorem first! For that, we need to build up some theory though... POSETS: A poset is a short word for 'partially ordered sets'. It is a set $P$ equipped with a binary relation $\leq$ that satisfies the following three conditions: $\forall a\in P$, $a\leq a$ (Reflexivity

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