Show that if a positive integer is odd, then it's square is odd.

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

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Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

Q: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: Find all possible digits that can be placed in the space so that the numbers is 1,182,1_4

A: Apply the rule for a number divisible by 9.

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

Q: Prove that τ(n) is odd if and only if n is a perfect square.

A: From the definition of τ(n), it represents the number of positive divisor of n. Let τ(n) be odd…

Q: Prove that there does does not exist a unique positive real number, x, with the property that x…

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Q: 1. Prove that the square root of 2 is not a rational number.

A: Definition of rational number : If x is a rational number then it can be written in the form of pq…

Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: Prove that there is no rational number whose square is 12.

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Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: a) The square of any integer is either of the form 3k or 3k + 1.

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that 3 is irrational number.

A: Rational Number: A number is of the form of pq, where p, q∈ℝ and gcd(p, q) = 1 Now we have to show…

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Let æ be a real number. Show that [3r] = [a] + æ + + 3

A: Here we have to show that 3x=x+x+13+x+23

Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Are the irrational numbers closed under multiplication?

A: Use closure property.

Q: Find a number that provides a counterexample to show that a given statement is false, For all number…

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Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

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Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: Use direct proof to show that product of two consecutive integers is even.

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Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove by contradiction that, if n is a positive integer which is not a perfect square, the sqrt(n)…

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Q: If the sum of any two integers is even, then so is their difference.

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

Q: Show that square root P is irrational where P is prime by using contradiction.

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Q: Prove that if n is a perfect square, then n +2 is not a perfect square.

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Q: Prove that the difference between the square of any odd integer and the integer itself is always an…

A: An odd integer is of the form 2n+1; n ∊ ℤ. Consider the difference between the square of an odd…

Q: The set of all positive multiples of v2 is countable.

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Q: Prove the given statement: The sum of an integer and its square is even.

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Q: Prove that there does not exist an irrational number whose square root is rational.

A: Since, (p/q)=p^2/q^2 is rational. The square of any rational is rational, hence no rational is the…

Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

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Q: show that () is an even number.

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Q: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.

Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: Show that every odd integer is the sum of two consecutive integers

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Q: Prove that there exists an irrational number whose square is rational.

A: We have to show that there exists an irrational number whose square is rational

Q: Show that 5 – V7 is an algebraic number.

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Q: Prove that 13,050.046 is not the square of a natural number.

A: Use the fact that an even number is perfect square the number is divisible by 4

Q: If a and b are odd positive integers, and the sum of the integers, less than a and greater than b,…

A: Let a and b are odd positive integers, and it is given that sum of integers, less than a and greater…

Q: Let x, Y be positive real numbers. If x2 – y² =1, then x or y (or both) is not an integer.

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Q: "There is no largest integer."

A: Given: To explain the given statement as follows,

Q: Show that 2 is a primitive root of 19.

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Q: How many integer solutions are there to 1 +X2+X3+X4+Xs=31 with

A: Given, x1+x2+x3+x4+x5=31 Number of Solutions when xi≥0 and xi>0.

Q: Prove that cube root of 2 is irrational by contradiction

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Q: Prove that there is not rational number whose square is prime

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Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.