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

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

Q: Prove the following statements: iii. Suppose that a, b, c and d are positive integers, prove that…

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Q: Prove that the following statement is true for all positive integers. 1 1 1 + + 4 8 1 2" – 1 + 2n Pn…

A: Given; Pn : 1/2 + 1/4 +1/8 +...+1/2n =(2n -1)/2n

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Prove or disprove: Let a and b be integers. If a divides b and a divides c, then a divides b-c.

A: Let a, b, and c are integers. Integer a is a factor of b and c.

Q: Let A = [0, 1] – Q. Determine if A is countable or uncountable. Prove your claim.

A: Let

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

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Q: Show that there exist no two positive integers nandr(r<n)such that "C, "C,+1 "C+2 and "C,.3 are in…

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Q: Prove that the following statement is true for all positive integers m and n:  m and n are…

A: m and n are multiples of each other if and only if m = n. for all positive integers m and n:

Q: Prove or disprove: For all integers a and b, if a+b is even, then a is even and b is even. Provide a…

A: Given statement is false because

Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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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: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: Let a and b be integers, not both zero. Thena and b are relatively primeif and only if

A: By standard theorem in number theory which states that:-

Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: a.) Let a and b are positive integers. Prove that (a,b) = 2(a/2, b/2).

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Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

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

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

Q: Prove or Disprove: If x and y are integers and x - y is odd, then x is odd or y is odd

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Q: Prove or disprove: For any integers a,b,c, if a { b and b X c, then a X

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Q: Let a and b be integers,not both zero .Then a and b are relatively prime if and only if

A: Solution are as follows

Q: Prove that if a and b are integers such that a|b, then either a = b or a = -b

A: Take a = 2 ,b = 4 . We have a divides b but neither a = b , nor a = -b.

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

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Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: Prove: | ab | = | a | * | b | for all a's and b's in the reals.

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Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

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Q: Prove using Direct Proof: For all integers a,b and c if a|b and b|c then a|c

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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: Let n, a, and b be positive integers such that ab = n. If (a, b) = 1, prove that there exist…

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Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…

Q: Prove or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

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Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: s) Let a and b be positive integers such that b< a < 10b. How many ways are there to place a…

A: In the given problem Balls are labelled and boxes are unlabeled. i.e. we have to distribute a…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n V3n

A: For all positive integers n, we need to prove that, 12·34···2n-12n&lt;13n

Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

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Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

Q: Prove that there do not exist positive integers a and n such that a? + 3 = 3".

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Q: Prove that A,(3, 2) = q² for any integer q>2. %3D

A: To prove : Aq3, 2=q2 for any integer q≥2 Pre-requisite : 1. Sphere-covering bound : For an integer…

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: Prove or disprove: For all integers a and b, if a +b is odd, then a is even and b is odd.

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Q: and

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

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Q: Prove that if a, b and c are positive integers such that a divides b and b divides c, then a divides…

A: It is given that a,b and c are three positive integers. It is also given that a divides b. Then it…

Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.

Q: Prove that if a is a positive integer, then 4 does not divide a2 + 5.

A: