An integer x is odd if x2 is odd. Let x, y and z be integers. If x y and y z, then x | z. For any integer z, if 3 divides z2, then 3 also divides z.

Q: If we want to prove by contradiction the statement: For any positive real number a there exists a…

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Q: shoew that r+E and  rE are irrational.

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Q: Given that U = {x : 1 ≤ x ≤ 10, x is an integer}, G = {x : x is a prime number}, H = {x : x is an…

A: Given that U = {x : 1 ≤ x ≤ 10, x is an integer},G = {x : x is a prime number},H = {x : x is an even…

Q: Prove the following: If ā and b are in R', then (5 -a)×(5 +a)=-2(āxb).

A: Given a⇀ and b⇀ are in R3. To prove :  b⇀-a⇀×b⇀+a⇀=-2a⇀×b⇀

Q: Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle and…

A: Given: a and m are length of sides of right triangle and m+1 is hypotenuse then a is an odd integer…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Let x, y ER. Ifx<y with x and y very close to each other. Then which of the following statements is…

A: Given x &lt; y

Q: If a, b e Z such that a b then the numbers a and b are called relatively prime numbers if gcd(a, b)…

A: Consider the given statement. We'll prove the statement by contradiction. Let's the numbers are not…

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

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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: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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Q: Let x and y be integers. Prove that if x2 + y2 is odd, then x+y is odd

A: Given x and y be an integers. We have to prove that if x2+y2 is odd then x+y is odd. Since given…

Q: (a) For all a, b E R, if a E Q \ {0} and b & Q then ; ¢ Q. (b) For all x, Y E R, if x is irrational…

A: a) Given: a∈ℚ\ 0, b∉ℚ To prove: ab∉ℚ b)  Given: x∈ℝ-ℚ, y∈ℝ-ℚ+  To prove: x+y∈ℝ\ℚ Rational Number: A…

Q: If a, b, and c are consecutive positive integers such that a < b < cand abc is NOT a multiple of 4,…

A: Given: a, b and c are consecutive positive integers. a&lt;b&lt;c  ……1   and, abc is not a multiple…

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

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Q: Let x,y,z∈ Z be integers. Prove that if x(y+z) is odd, then x is odd and at least one of y or z is…

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Q: If 90% of x is 8100 then x is

A: Given,         90% of x is 8100.

Q: For every pair of integers x and y, if x – y is odd, then x is odd or y is odd.

A: Let x-is even integer and y-is even integer. x=2m, y=2n

Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

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Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: If α is even, prove that α-1 is even. If α is odd, prove that α-1 is odd.

A: We know that the multiplication of 2-cycles can help us find any permutation of a finite set, so we…

Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: For a positive integer i, let 3 = |2,3+ i 3 A¡ = {x €R 2<x< 3 + There are finitely many elements in…

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Q: For any positive integers m, n (

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Q: For each of the values on n and d given, find integers q and r such that n = dq + r and 0 is less…

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Q: If x is even and y is odd, show that (x + y)? is odd.

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Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

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Q: Let x and y be integers. Prove by the contrapositive that if x2 - y2 is even, the x + y is even

A: Let x and y be integers such that x2-y2 is even. We have to prove that x+y is even. Suppose that x+y…

Q: For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m- 1.

A: Given that m is an integer and x is a real number which is not an integer. we know that for any real…

Q: Prove by contradiction. If a and b are rational numbers, b ≠ 0, and r is an irrational number, then…

A: Assume that the number a + b r is rational. Since a and b are rational, then a and b can be…

Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

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Q: a. {x]x 54, x € even integers} d. {x]x > 54, x €negative integers} O {x]x 54, x Enegative integers}…

A: Set builder form of the given set

Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: There exists a real number R which is greater than 5 O R= {7, 42,2t, and 97/3) O R= {1, 2, and T/6}

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Q: For any integer m and any real number x, if x is integer, then x]+[m-x] = m- 1. -. not an

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Q: B. Prove the following by contradiction: 1. If 3x + 2 is odd, then x is odd. 2. If there are two…

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

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

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: -) If A and Bare 3 x 3 such that det A = 3 and det B = 2 } Find, if possible, the value of det(A²…

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Q: Suppose a, b, c and d are integers. Prove that  if a divides b and c divides d, then ac divides bd

A: According to the given information:   It is given that a divides b, there is an integer x such that…

Q: If there exist integers r and s such that ra + sb = d. then d = (a, b). %3D

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Q: If A = {x E N x is odd, x <,7pw many SUBSETS of Acontain no even numbers?

A: If a set contains ‘n’ elements, then the number of subsets of the set is 2n.

Q: tet r and m be integer, prove that 17/20+3m iff, 17/9n

A: Solution

Q: If a ≠ 0, the degree of axn is__________.

A: We have to fill in the blanks: If a≠0, the degree of axn is ----. We know that degree of any…

Q: Find B and A such that o(x) is continous: o(x)= {13-Bx.     x<-5}           {3x^2 -1.   -5 less…

A: The given piecewise function is: ox=13-Bx      for  x&lt;-53x2-1        for   -5≤x&lt;42x+A…

Q: If a and b are two integers such that a = bq +r, with b> 0, then %3D gcd(a, b) # gcd(b,r) gcd(a, b)…

A: Let a = bq + r   … (1)Now,  let d1   = gcd ( a,b) then ⇒d1  | a  and   d1  | b ⇒d1 | a - bq   and d1…

Q: Prove that if a is odd and b is odd, then (a + 1) b is even

A: Number system.. properties