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Tuesday , August 22 2017

## For n ≥ 1, established that the integer n(7n^2 + 5) is of the form 6k

Solution: According to division algorithm, we have n = 6q + r ; 0 ≤ r < 6 Let, A = n(7n2 + 5) For r = 0, we have A = n(7n2 + 5) = (6q + r) {7(6q + r)2 + 5} = 6q (7.36q2 + 5) = ...

## Verify that if an integer is simultaneously a square and a cube, then it must be either of the form 7k or 7k + 1

Proof: Let A be an integer. We have to show that the cube and square of A is of the form 7k or 7k + 1. First we show that, a square is either of the form 7k or 7k + 1. According to the division algorithm, we have A ...

## PDF|SOLUTION OF EXERCISE – 9.2| CLASS – 7

PDF|SOLUTION OF EXERCISE – 9.2| CLASS – 7

## Ideals | Lattices and Boolean algebra

Definition: A non empty subset I of a lattice L is called an ideal of L if (i) a, b ∊I ⟹a ˅ b ∊ I (ii) a ∊ I, l ∊ L ⟹ a ˄ l ∊ I. Example: Let {1, 2, 5, 10} be a lattice of factors ...

## Basic properties of congruence and their proved

i) a ≡ b (mod n) Proof: For any integer a, we have 0/n = 0 ⟹ (a – a) / n = 0 ⟹ a – a = 0.n ⟹ a ≡ a (mod n) (proved)    ii) If a ≡ b (mod n) then b ≡ a(mod n) ...

## Congruence

Definition: Let n be a fixed positive integer. Two integers a and b are said to be congruent modulo n, symbolized by a ≡ b (mod n) if divides the difference a – b ; that is, provided that a – b = kn for some integer k. a is ...

## There are an infinite number of primes of the form 4n + 3.

Lemma: The product of two or more integers of the form 4n + 1 is of the same form. Proof: It is sufficient to consider the product of just two integers. Let us take k = 4n + 1 and k′ = 4m + 1. Multiplying these together, we obtain ...

## Sublattice and Convex sublattice

Sublattice: Let (L, ˄, ˅) be a lattice. A non empty subset S of L is called a Sublattice of L if S itself is a lattice under same operations ˄ and ˅ in L. Or, A non empty subset S of a lattice L is called a Sublattice of ...