(a) If m, n are odd integers, then mn is an odd integer. (b) If m, n are integers and mn is even, then m is even or n is even. (c) If z, y are real numbers and x +y is irrational, then z is irrational or y is irrational.

Q: 1. Show that the square root of 3 is irrational as follows: (a) If 3p² = q² for p, q E Z, show that…

A: Let us prove the above result.

Q: If m and n are integers and m x n = 0, then what is true for the values of m and n?

A: If m and n are integers and m×n=0

Q: Find, with proof, all integer solutions to 15s + 27t = 6.

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Q: (1) Give a proof by contradiction of the statement: If n? + 3n – 5 is even, then n is odd.

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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: 7. Suppose a1, a2,...,ɑn are integers, r is a solution to the equation 1" + a,r"-1 + a2r"-² + + a, =…

A:

Q: How many non-negative integer solutions (O is allowed) are there to r +y+ z = 24?

A: The given equation is, x+y+z= 24 We know that, For equation,  x1+x2+......xr=n  There are n+r-1Cr-1…

Q: 3. Which of the following statements are true? (10 points)(a) If x is rational and y is irrational,…

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Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: Give a contrapositive proof of the following: If n is an odd integer, then the equation x2+ x -n=0…

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Q: you’re thinking of a number 1-50 and i have to guess what it is. i asked these questions in this…

A: Given: A numbers between 1-50. To find: Find a number greater than 25, divisible by 2, 3 and 5 and…

Q: 5. Find all integer solutions to 6s+ 14t+21u = 1.

A:

Q: are irrational. Prove that √11-√√7 is irrational." Prove indirectly: "It is known that √7 and √11 a

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Q: Prove that √2 + √7 is an irrational number

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Q: Ex For this problem, you may assume any previous results from class as well as the known fact that,…

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Q: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

Q: d) How many non-negative integer solutions are there to the following problem? I1 + x2 + I3 = 10…

A: The problems related to calculating the number of ways can be solved using the combinations. The…

Q: Find all integer solutions to 25s+ 14t = 1. O

A: Given equation25s + 14t = 1We have to find all integer solutions for s and t.

Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Which of the following statements is true? a) If an integer n is odd, then n2 + 2n is even. b) If an…

A: (a) We need to know whether n2+2nis odd or even when n is odd. If n is then odd n=2k+1 for some k∈ℕ.…

Q: 10. Prove that the following are irrational numbers: (1) √3 (ii) 1 + √2 (iii) √3+ √2 (iv)

A:

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: 7. Let y be irrational and x ‡ 0 a rational number. Show that the product xy is irrational.

A: We will prove given statement by method of contradiction.

Q: . Classify as rational or irrational. • 8/9 • √3/4 • 49 • √6 •³√8

A: As rational numbers are those numbers which can be written in the form ho p/q where p and q are…

Q: (4) Consider the statement "There exists an even integer n, such that for all rational number q, we…

A: Statement "There exists an even integer n, such that for all rational number q, we have n²>q"

Q: Prove by contradiction that if 3x? 2 is irrational then r is irrational.

A: We have to show that x is irrational if 3x2-25 is irrational. We have to prove the statement by…

Q: 1. For any integer n, the quantity 3 (n² + 2n + 3) - 2n² is a perfect square.

A: We need to show that for any integer n following quantity is a perfect square: 3(n2+2n+3)-2n2

Q: The principle argument of a complex number, denoted by Arg(z) is the argument of z which is…

A: For complex number x +iy  1st we will find α using tan α =yx Then depends on quadrant principle…

Q: Let n be a real number. Prove that if n is irrational, then 5n - 1 is also irrational.

A: Given that n is real number. If n is irrational number then we need to prove that 5n-1 is also a…

Q: Denote by an the nth positive integer that is not a perfect square. For example, we have…

A:

Q: Given the following statement (M): “If b is an irrational number then r = is irrational." None of…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Q. No.3 (a) Show that v5 is an irrational number. (b) Let x and y are two irrational numbers. Then…

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Q: How do I prove that if a,b are rational numbers, then so is (a+b)/2

A: If a,b are two rational numbers, then we have to prove thata+b2 is also a rational number.

Q: 3. Prove or disprove: there exist rational numbers a and b such that a/b is irrational.

A: Suppose there exist rational numbers a and b such that a/b is irrational.