## What is Feasible Region in solving LPP and what are its associated terms?

In my first blog on Linear Programming, you got an idea of Linear Programming and Linear Programming Problems (LPP). In my second blog of the series, you learned about some basic terms of LPP. In the present blog, I move ahead in giving you the beginning of (graphical) technique of solving an LPP, and you must know some more basic terms which are associated with finding of solution of an LPP using graphical method. We go back to our first problem which I gave in my first blog

## What is Optimal Feasible Region in solving LPP, and how to solve an LPP having feasible region in th

In my first three blogs on Linear Programming, I gave you an idea of Linear Programming Problems (LPP), of some basic terms of LPP and of the terms to be used in (graphical) technique of solving an LPP. In this blog, the final lesson on the understanding of solving LPP shall be taken up and then we shall do questions. In the previous blog, you understood the feasible region and feasible solutions. Of course, there are an infinite number of feasible solutions available in the

## Some terms of Linear Programming you must understand before you begin

In my previous blog, I was a bit cautions in not giving you a big bout of some Latin and French type of taxonomy of Linear Programming and restrained myself in focusing only on giving you some basic understanding in most layman kind of way on what is Linear Programming. Since, you have already understood what Linear Programming is, we shall gradually understand various terms associated with it. The first and core word is “Optimization”. All LPPs are optimization problems as e

## What is Linear Programming?

The word “Linear Programming” is made up of two words, linear and programming. The word ‘linear’ means we are getting a linear equation in formulation of some problem. The ‘linear’ means that the equation is of single degree in one or two variables as an equation of one or two variables in single degree in two-dimensional geometry will always give you a straight line (hence linear). The word ‘programming’ gives us an impression that the solution shall involve mathematical tec

## An observation regarding two APs with same common difference

For two APs (Arithmetic Progressions), with initial terms being a and b respectively, and common difference being same, say d, the difference between their nth terms remains same, and is equal to a – b, as, {a + (n – 1)d} – {b + (n – 1)d} = a-b, the constant. I do a question. Two APs have the same common difference. The difference between their 100th terms is 100. What is the difference between their 1000th terms? Given, T2(100) – T1(100) = 100; i.e. {a + (100-1)d} – {b + (10

## The nth term of an AP is always a linear explanation

This was a good question from Arithmetic Progression which I thought that not much brooding on how to solve it was necessary at all. The question is: Justify if it is true to say that the following are nth terms of an AP. (i) 2n – 3 (ii) 3n^2 + 5 (iii) 1+n+n^2 I found that that a linear explanation of nth term, like pn+q where p and q are rational numbers can always be put in the form: (q+p) + (n-1)p; which is nth term of an AP with initial term being (q+p) and com

## Remainder, you are awesome (2) ....

Today I happened to have an encounter with the same type of question that I had had in my blog. The present blog is the extension of previous blog on the same issue. The question is: A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainder 1, 2, 3, 4, 5 respectively. How many integers between 0-100 belong to set? A. 0 B. 1 C. 2 D. 3 I solved it like this: Let the number be P. Then, as given: P = 1 mod 2, i.e. P = -1 mod 2 P = 2 mod 3, i.e

## Remainder, you are awesome ....

Remainders are wonderful and you can do wonders with remainders in solving some otherwise looking dreadful questions of Real Numbers. See this example. Suppose, you divide 15 by 7, you have remainder 1, and you can always write it in terms of Euclid Division Algorithm, as: 15 = (2x7) + 1. Though the remainder is always a positive integer, you can always think it writing like this too: 15 = (3x7) + (-6). And this can be written as: 15 == -6 mod 7. This idea is wonderful in sol

## Find the remainder when a natural number n satisfying following conditions is divided by 5:

The conditions are: What remainder does n give when divided by 5? First congruence relation gives, n = 2 or 3. Second congruence relation gives, n = 3. Thus the remainder when 3 is divided by 5, is 3. Thus n = 3 is the answer.

## What is the remainder when the following sum is divided by 4?

The sum is: Sol: The beauty of the solution lies in application of Modular Arithmetic. it is a simple question if the student has some good knack in Modular Arithmetic. The equivalence class mod 4 has four integer members 0, 1, 2, 3. The member 0 will belong to set {4, 8, 12, ….,96, 100}, as all these number when divided by 4 will give remainder equal to 0. Hence, all these numbers {4, 8, 12, ….,96, 100}, when raised to power 5 and divided by 4 gives remainder equal to 0, as

## A problem offering a solution

Question: If a, a + 2 and a + 4 are prime numbers, what are the number of possible solutions for a? (1) one (2) two (3) three (4) more than three. The question appeared in CAT 2003, Re-test. The question belonged to Number Theory, and it suited me best to use Modular Arithmetic in solving it. Believe me, before I started I had not thought of its use, but when I saw that three consecutive members of an AP with a common difference of 2 were divisible either by 2 or by 3, with t

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