An inequality for positive integers Show that if a, b, c, and d are positive integers, then (a? + 1)(b + 1)(² + 1)(d² + 1) abcd 2 16.

Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m - x] = m-1.

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Q: Using indirect proof:  Let m be an integer. Prove that if m is odd then m² is odd.

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Q: Let R be the set of 50 distinct positive integers. These integers are divided by 23. Show that at…

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Q: A = B? 7. In each of the following cases determine whether A ⊂ B, A = B or A ? B. Give reasons for…

A: Given two sets, one set is called a subset of the other if all the elements of that set are present…

Q: Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing…

A: Since you have asked multiple questions, we will solve the 1st question for you. If you want any…

Q: Proof that 3√2 is an irrational.

A: Given query is to prove that 3√2 is an irrational.

Q: . Counterexample. Find a counterexample to show that the following conjecture is false. a. For all…

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Q: EVALUATION: Prove the following. 1. Using direct proof, show that "if x and y are sum of two integer…

A: Note: Since you have posted multiple questions we will solve the first question for you. If you want…

Q: 27. Proving a Factorization Show that x - y is a factor of x" - y" for all natural numbers n. [Hint:…

A: To prove(x-y) is a factor of xn-yn for all natural number n.

Q: 3. Invent two interesting and original positive integers a and b. Let f(x) = ax +3. (a) (b) What is…

A: Given function is fx=ax+3. To find:  a  f10   (b). A number x such that fx=b Solution: Given…

Q: 2. Using mathematical induction prove the formula: For every real number r except 1, and any integer…

A: Given:  The formula: for every real number r except 1 and for any integer n≥0, ∑i=0nri=rn+1-1r-1, To…

Q: Step by step else downvote, with clear handwriting. Prove that for any real number x, there exits a…

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Q: Number Theory: 10) Find the primes for which 11 is a quadratic residue.

A: To find all primes ’p’ for which 11 is a quadratic residue. This means,

Q: Axiomatic ProofsProve a*(b-c)=a*b - a*c for all integers a, b, and c

A: The below expression is known as distributive property: -a*(b – c) = (a*b) – (a*c) for all integers…

Q: (3і - i = 1 2 (3i – 2) = n(3n - 1) 2

A: By using the principle of mathematical induction prove that  ∑i=1 n(3i-2)=n3n-12

Q: Discrete mathematics: Prove that if a is an integer other than 0, then a divides 0.

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Q: Gaussian integers has the same strength as the set of natural numbers N.

A: We need to use the fact that Gaussian integers is isomorphic to Z×Z

Q: Integers a and b are such that (a+3-5)² + a- b,5 = 51. Find the possible values of a and the…

A: Note: It is assumed that the equation is a+352+a-b5=51.   It is known that x+y2=x2+2xy+y2.

Q: USING DIRECT PROOF Let m be an integer. Prove that if m is odd then m² is odd

A: topic - numbers

Q: By considering uniqueness of factorization of integers, find all pairs (x, y) ∈ N^2 such that x^2 −…

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Q: gcd(a, 1) = 1 for any integer

A: True.

Q: 2. Even and Odd (a) Use a direct proof to show that the sum of two odd integers is even. (b) and n,…

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Q: Prove  that for any integer k, if 3k+1 is even ,then k is odd. State the method of proof used.

A: *** As per our guidelines, we are supposed to answer only one question at a time. Since you have…

Q: Methods of Proof: Prove the following using the methods of proof indicated. 18. If x is an odd…

A: As per the company guidelines We can first question only. Please understand. Kindly repost the…

Q: Using Contrapositive Prove that for any integer a, the integer a(a+1) is even

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Q: Þrove that for all positive integers n, 1° + 2° + 3° + . + n° = (ma+)". a. Show it is true for n= 1.…

A: The following two steps are used to prove a statement P(n) by the principle of mathematical…

Q: 2. Even and Odd (a) Use a direct proof to show that the sum of two odd integers is even. Use an…

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Q: using proof of contradiction show that if n is any odd number and m is any even integer then 3^(n^3)…

A: Let n be any odd number and m be any even integer. To show 3n3+2m2 is odd. Suppose not. Then 3n3+2m2…

Q: Cant seem to figure this proof out

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Q: (a) Use proof by contradiction to show that there is no positive real number r such that x-2 X > X…

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Q: GCD between two integers is the GCD between the divisor and the remainder of the division. True or…

A: True; because GCD defined as the Greatest Common Divisor between two numbers is calculated by the…

Q: 9) The dimensions of a gift box are consecutive positive integers such that the height is the least…

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Q: sum of p¨ powers of n¨ roots of unity is zero ,if p is not a n-1 multiple of n .Proof n" roots of…

A: We will prove this using, sum of nth roots of unity and check that it's zero or not. if it's zero…

Q: * mathematician wishes to prove the following statement about the integer x. Statement (1): If x² is…

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Q: 1. Using exclusively the axiomatic description of real numbers enclosed (see at the end of this…

A: As per our guidelines I am solving only 1st three for others kindly resubmit the question mentioning…

Q: Question Prove that for all positive integers a and d, if d|a then d is less than or equal to a.

A: Solve the problem

Q: Using indirect proof, prove:  Let a, b, and c be integers. Prove that if a divides b and a divides…

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Q: TRUE OR FALSE  FOR ALL x, x is any INTEGER: x^2 + 2x + 1 = 0. *

A: Here we have to prove or disprove,     That for all x , x is an integer       Given x2 + 2x + 1=0

Q: Question A positive integer is 7 less than another. If 3 times the reciprocal of the smaller integer…

A: We have to solve

Q: An algebraic number is any real root of an equation ao + a₁x + +anx"=0, where the coefficients a;…

A: We use properties of Countable union of countable sets and set of polynomials.

Q: 03. a) Use divided differences and interpolation to obtain the formula for the sum of the first n…

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Q: You are constructing a box from a piece of cardboard with dimensions 3 by 5 meters. You cut…

A: Since you have asked multiple question As per our policy, we will solve the first question for you.…

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c,  then a divides (b-c).

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Q: 5. (10 points) Prove that for all positive real numbers r and y, one has the inequality > 2 (u+ 112…

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Q: 1.Prove that the product of two consecutive integers is divisible by 2.                   (Use…

A: Product of two consecutive integers is divisible by 2

Q: Donald's "Theorem" states that if x and y are even integers, then ry must be a perfect square. (Note…

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Q: True or False? 5. Let x and y be integers. If 5x − 7 is even, then x is odd. 6. Let x and y be real…

A: To find which of the above is true or false :-

Q: Prove using proof of contradiction. Include explanation.   If n is an even integer, then 7n + 4 is…

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