a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

Q: Prove that for all positive integers n, 1 3 2n – 1 1 X...X 4 V3n 2n HINT: You will find it difficult…

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Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: antip . Prove that 4" -3"-7 is a multiple of 84 for each positive even m.

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Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: 3. Prove that √12 is irrational using the Fundamental Theorem of Arithmetic.

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Q: Prove or Disprove the following statements. You may type or hand write your answer. a. For all real…

A: The answer is given as follows:

Q: Prove by Contradiction: There are no integers x and y such that = 4y+2.

A: We have to prove the statement "There are no integers x and y such that x2=4y+2" by contradiction.…

Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: Use proof by contradiction to show that 2 is not a rational number. Note that you may use the…

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Q: a. If b is any nonzero real number, prove that |1/b| = 1/| b | . b. Prove that |a /b | = |a|/| b|…

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Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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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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Q: Prove that there is no rational number x such that x2= 3.

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Q: Prove that for integers a and b, (a -3)b2 is even if and only if a is odd or b is even.

A: We need to prove that for integers a and b, (a -3)b^2 is even if and only if a is odd or b is even.

Q: Disprove this statement by giving a counterexample. For all nonzero real numbers, a, b, c, and d,…

A: Here a,b,c,d all are non zero real .So (a/b) and (c/d) are defined.  Let,  a = 1 , b = 2 , c = 3 , d…

Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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