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Prove that if x + y is even, then the product xy(x + y)(x - y) is divisible by 24, and that without this restriction, 4xy(x- y)(x + y) is divisible by 24.
Explain the main differences among integers, rational numbers, real numbers, and irrational numbers. How are these used in everyday life?
Applications of a System of Equations : Solving A Word Problem.Tickets for a concert was sold to adults for $3 and to students for $2.
The file was set with 11 buckets numbered 0 to 10, where each bucket can contain at most 3 records.
Perturbed Linear System.Consider the perturbed linear system
Solutions of the systems of congruences.Find all solutions of each of the systems of congruences:-
Show that 11 divides 10a+b if and only if 11 divide a - b. Use this to show that 11 divides 232595.
Linear independence of vectors &Linear dependence of vectors. Linear dependence of vectors. Linear independence of vectors
Proofs regarding Fermat numbers.The Fermat numbers are numbers of the form 2 ^2n + 1 = Fn . Prove that if n < m , then Fn ¦? m - 2.
What does f: Z----> Z or f: A--->A or f: N--->N mean? Are these the set of integers, natural numbers or any set in general (it is not the complex set)?
The general solution of the linear Diophantine equations.Find the general solution
Systems of Equations : Matrices.Please give step by step instructions and name each step like triangular form
Let f:N x N -> N be the function defined recursively as follows Use induction on the sum i + j to prove that f(i, j) = 2i + j + 6 for all (i, j) in N x N.
Suppose that you need to take a trip from city 1 to city 9 in below figure in the least amount of time.
Repeat the preceding part after interchanging the order of the two equations.
A classic unsolved problem in number theory asks if there are infinitely many pairs of `twin primes', pairs of primes separated by 2, such as 3 and 5.
Let m be the least common multiple of a and b, and let c be a common multiple of a and b. Show that m divides c.
Knowing that {Hn}, n = 0, 1, 2... are orthogonal and with respect to a suitable weight function p(x), write down first two orthonormal Hermite polynomials.
After eating at the restaurant, keith, Sandy, and Jason decided to divide the bill evenly. If each person paid 26 dollars, what was the total of the bill?
Explain in your own words your understanding of the concept of slope as it relates to graphs of linear equations. Illustrate with examples
Why do intersecting lines represent a unique solution? Give examples to support your answer.
The aim of this exercise is to prove that the Mordell equation y2 = x3 - 5 has no solutions. We proceed by contradiction and assume that (x, y) is an integral.
Solve by substitution or elimination method:A university bookstore recently sold a wire bound graph-paper notebook for $2.50
Prove that if p is a prime number of the form 4n + 1 , then the following congruence holds,x2 =- 1mod p
Proof of diagonalizability.If A is diagonalizable and B is similar to A then B is also diagonalizable.