## An easier way to find out the unit digit of a number indexed with a positive integer using Modular A

In many of competitive exams, you are asked to find out unit digit in some n^p where n and p are positive integers. Sometimes it is a lengthy process to find out the correct answer, and remember one principle, avoid unnecessary multiplications. Hence given below is the method using Modular Arithmetic where less and less calculations have been used. Let us begin with an example. What is the unit digit in 2^103? If you do it with traditional method, you know that the unit digit

## Algebra of Congruent Modulo n

For any integers a, b, c, d and positive integer n a ≡ a (mod n) If a ≡ b (mod n), then b ≡ a (mod n). If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n). If a ≡ b (mod n) and c ≡ d (mod n), then a ± c ≡ b ± d (mod n). If a ≡ b (mod n) and c ≡ d (mod n), then ac ≡ bd (mod n). The division in two congruent modulo n is not always possible, means, one cannot say as a rule that If a ≡ b (mod n) and c ≡ d (mod n), then a/c ≡ (b/d) (mod n). Hence there is a concept of inverse c

## What is Ratio

Consider this example. In a mixture of 35 liters of milk and water, the milk has four parts and water has part. Obviously there are a total of 5 parts; hence one part is equal to 7 liters. The quantity of milk being 4 parts i.e. is 28 liter and that of water is one part, i.e. 7 liter. We shall say that the ratio of milk and water in this solution is 4 and 1. This shall be written as 4:1. Mathematically, this is 4/1. Hence, A:B, i.e. A/B is a ratio between two quantities of sa

## Equivalence Class in Congruent Modulo

Can you the think the set of remainders as a class? See the example. Let a == b mod 2. Every integer is congruent modulo 2 to exactly either 0 or 1. For example, 17 when divided by 2 will give you remainder 1 and 24 when divided by 2 will give you remainder 0. In fact, every integer when divided by 2 will give you remainder either 0 or 1. Hence all integers on number line shall be grouped in two classes of numbers wrapped up around 0 or 1. These classes are known equivalence

## Modular Arithmetic (Congruent Modulo n)

Play with remainders What a superb idea did Gauss have long time ago!!!! If you add two numbers, the remainders obtained after dividing the two numbers also get added up. For example: Similarly, if you subtract two numbers, the remainders obtained after dividing the two numbers also get subtracted. For example: Similarly, if you multiply two numbers, the remainders obtained after dividing the two numbers also get multiplied. For example: The same relation is not ALWAYS possib

## A modest thought on CBSE paper of XII Mathematics 2016

I am not a teacher of Mathematics in any school and I don’t teach in any institute either, nor do I have any coaching institute. This has been written by me simply to share my views as a common man on the gradual and perpetual deterioration in studies of Mathematics. Basically certain segments of this paper are addressed to students of XI-XII (with Mathematics) and hence I have divided this article in two parts, one, about the CBSE paper of XII Mathematics, 2016, and two, abo

## What are THEY teaching?

What are THEY teaching? Alas! To my astonishment, in many supplementary books of Mathematics I see the most tedious methods of solving even the easiest problems. In one (so very famous??) supplementary book of Maths XI, I found that the author was explaining the two forms of ellipse (One having major axis along X axis and another having major axis along Y axis) in most ridicules way, where this book says: always make ‘a’ and ‘b’ the denominators of X and Y respectively for bo

## Why do I love Mathematics?

Why do I love Mathematics? Doing Mathematics is fun, thrill and excitement. I don’t know why the teachers don’t make themselves belong to the subject. Why don’t they open before the students the euphoria Mathematics possesses? It is only Mathematics that helped human race to achieve greater insights into the enigma of nature. Could we expect happening of various discoveries of Physics, both Mechanical (Newtonian) and Quantum, without deploying the means of differential and in

## CLAT प्रश्नपत्र में सफलता के लिये गणित की भूमिका

देश के 15 प्रतिष्ठित लॉ स्कूल्स के लिये प्रतिवर्ष आयोजित होने वाली क्लेट (CLAT – Common Law Admission Test) परीक्षा में 200 मार्क्स के पेपर 200 ऑब्जेक्टिव प्रकार के प्रश्नों में निम्नानुसार विषय समाहित होते हैं: English including Comprehension: 40 Marks General Knowledge and Current Affairs: 50 Marks Elementary Mathematics (Numerical Ability): 20 marks Legal Aptitude:

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