(a) Use proof by contradiction to show that there is no positive real number r such that x-2 X > X x+2 (b) Use proof by contraposition to prove that the following statement is true for all positive integers m, n: if the product mn is odd, then both m and n are odd.

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Prove or disprove: There exists an integer a such that 14 | (3a — 5) and 21 | (2a + 3).

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: Use a proof by contradiction or by cases to show: No integers y and z exist for which 2y2 + z2 = 14

A: Result: Product of two positive term is always positive and product of two negative term is always…

Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

Q: Prove or give a counterexample to the following statement. For each positive integer a, there exists…

A: Given: 12b2+b<1ab2To check whether the given inequality is true or false keeping in mind for each…

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

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Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

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Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

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Q: Prove that 2 + v3 is irrational.

A: Given: To prove: 2+3 is irrational.

Q: Prove that the product of two integers, one of the form 3k₁ +2 and the other of the form 3k2 +2,…

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Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

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Q: Use a direct proof technique to prove the following theorems: For all integers r and y, r + y - 3 is…

A: We have to prove that , For all integers x and y , x2+y2-3 is not divisible by 4. We will have 3…

Q: Prove that the roots of X2+(1-K)x+k-3=0 are real for all values of k?

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Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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Q: Find all positive integers n that is a product of two distinct primes such that o(n) = 60.

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Q: ove or disprove: There exists an integer a such that 14 (3a-5) and 21 (2a +

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Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: d) The product of two numbers, a and b is zero if and only if at least one of them is zero; e) For…

A: We have to show that product ab=0 iff at

Q: = There do not exist integers x and y such that 15x + 1 There is no integer n such that n² + 1 is…

A: Assume the result is true then obtain a contradiction.

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Use a suitable method to prove or disprove the following statement formally and precisely: If the…

A: Even and odd numbers can also be characterized as: Even Number: A number that is divisible by 2,…

Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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Q: (1) Disprove the statement "For all integers a, b if a | b2 and a < b then a | b." by giving a…

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