Gauss called it the 'fundamental theorem' and published 6 proofs of it. Since then quadratic reciprocity has been an obsession of the mathematical community. Over 200 proofs has been published. I encountered a very simple and elegant proof. Here is a pdf file with a simple 2-page proof. quadratic reciprocity
This is an objective problem 151 from TOMATO based on Consecutive composites, useful for Indian Statistical Institute Entrance Exam. Let $n = 51! + 1$. Then the number of primes among $n+1, n+2, ... , n+50$ is (A) $0$; (B) $1$; (C) $2$; (D) more than $2$; Discussion: $51!$ is divisible by $2, 3,... 51$. […]
If three prime numbers, all greater than $3$, are in A.P. , then their common difference (A) must be divisible by $2$ but not necessarily by $3$; (B) must be divisible by $3$ but not necessarily by $2$; (C) must be divisible by both $2$ and $3$; (D) need not be divisible by any of […]
Let N be a positive integer not equal to 1. Then note that none of the numbers 2, 3, ... , N is a divisor of (N! -1). From this we can conclude that: (A) (N! - 1) is a prime number; (B) at least one of the numbers N+1 , N+2 , ...., N! […]
The number $1000! = 1.2.3...1000$ ends exactly with (A) $249$ zeroes; (B) $250$ zeroes; (C) $240$ zeroes; (D) $200$ zeroes; Discussion: To find the number of zeroes at the end of n! we just need to figure out the number of 5's occurring in prime factorization of it. Why? Because there are much more 2's […]
This post consists of Problems and solutions from TIFR 2013 Paper. Try to solve them and then read their solutions. TIFR 2013 Paper PART A (Linear and Abstract Algebra) Problem 1 Problem 2 - Automorphism of the Additive Group of Rationals Problem 3 - Existence of Real Root Problem 4 - Existence of Complex Root […]
This is a Geometry theorem based on Angles adding up to 180 degrees. It is helpful for Mathematics Olympiad. Try to prove the statement! Statement: Angles adding up to 180 degrees ABC be an isosceles triangle with AB = AC. P be a point inside the triangle such that, $ \angle ABP = \angle BCP […]
Every subgroup of order 74 in a group of order 148 is normal Discussion: True We will prove a much general claim: if index of a subgroup is 2, then that subgroup must be normal. Suppose $ H \le G $ and $ [G:H] = 2$ . Now, if $ g \in H $ then […]
This is an (ever-growing and ever-changing) list of books, useful for school and college mathematics students. If you are working toward Math Olympiad, I.S.I., C.M.I. entrance programs or intense college mathematics, these books may prove to be your best friend. If you are taking a Cheenta Advanced Math Program, chances are that you will referred […]
Let's discuss this objective problem number 14 from ISI BStat BMath. Try to solve the problem and then read their solution. Problem 14 f(x) = tan(sinx) (x > 0) To understand the graph of a function, easiest and the most proper method is to apply techniques from calculus. We will quickly compute, derivative and second […]