6. Use a direct proof to show that the product of two odd numbers is odd. 8. Prove that if n is a perfect square, then n + 2 is not a perifect square 12. Prove or disprove that the product of a nonzero rational number and an irrational number is irrational.

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A: We have to check

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A: To show that if a positive integer is odd then its square is odd

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A: Show 2.526262626.... is a rational number by expressing it as a ratio of two integers as follows.

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A: Since you have posted a multiple question according to guildlines I will solve first question for…

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Q: Which statement is not correct? Sum of two irrational numbers is irrational. O For every integer n,…

A: The incorrect statement is : sum of two irrational numbers is irrational.   the detailed solution is…

Q: Are the irrational numbers closed under multiplication?

A: Use closure property.

Q: 9. Prove that V2 is an irrational number.

A:

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A: To prove the given statement by using mathematical induction.

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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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A: We have been given that nth root of b exists where b is the real number.

Q: Prove: The product of an irrational number and a nonzero rational number is irrational.

A:

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Q: Show that 3 + V5 is irrational number

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

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

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Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

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Q: Abby notices that for the first twenty perfect squares, each square is either a multiple of 5, one…

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Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

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Q: 3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+ 6. Hint: Recall…

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Q: 3. Prove that if n is a triangular number, then so are 9n +1, 25n + 3, and 49n + 6. Hint: Recall…

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Q: 5. What is the completion of (X, d). where X is the set of all rational numbers and d(xr, y)

A: