7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... + an = 0. Prove that ifr is rational, then it is an integer. 8. Prove that for every n > 1, the sum 1 1 1 + + ... 2 is not an integer.

Q: How many integer solutions are there to the equation with x₁ > 1, 2 > 2, x3 > 2, x4 > 0 ? X1 + x₂ +…

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Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

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Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

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Q: write down the proof of the statement " √2 + √5 is not rational" in latex

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Q: (a) How many nonnegative integer solutions are there to the equation ₁ + x₂ + x3 + x₁ = a? (b) How…

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Q: 5. Consider the equation x₁ + x₂ + x3 = 18, where 2₁, 22, and 23 are integers. How many solutions…

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Q: 6. Prove that the statement "If x is a real number such that x² > 4 then x > 2" is false.

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Q: Find the maximum value of a, x, +a,x, + .+ax to the condition that x +x, +. + x =1 for given…

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

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Q: 3. Show that √2+√3 is irrational. Show that √n-1+√n+1 is irrational for every integer n ≥ 1.

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Q: a. Find all pairs of integers such that one is a multiple of 7 and the other is a multiple of 11 and…

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Q: Prove that if x is a rational number andy is an irrational number, then x + y is an irrational…

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Q: (b) Prove that 27 (2⁹⁹ +1).

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

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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: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: 3. In the domain of real numbers, explain what the following statement says in English. A rational…

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Q: 3x²³-2x² + 3kx + 18 by x- 6 to be equal to zero, k must be equal to.

A: Use division

Q: 2 4. Express 989 = 23 43 in the form x² + xy + 5y².

A: The given number is 989. The objective is to express the given number in the form of x2+xy+5y2.

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Let a, b, c be positive integers such that a is a multiple of c and a2 + b2 is not a multiple of c.…

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Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: 5. Let m be the smallest positive integer such that m² + (m + 1)² ++ (m + 10)² is the square of a…

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Q: 3. Prove that the sum of any two rational numbers is a rational number

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Q: 43. Show that there cannot exist rational numbers a, b such that V3 = a+b/2.

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Q: For any integers a and b, a + b ≥ 15 implies that a ≥ 8 or b ≥ 8.

A: Let a+b≥15, where a,b are integers.   Case 1: b<8 Then: b<8⇒-b>-8⇒15-b>15-8⇒a≥15-b>7…

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

A: Suppose a|b and a|c. Then there are integers m and n such thatb = am and c = an.Then b + c = am + an…

Q: 6. Prove that x is a rational number if and only if 2x - 1 is a rational number.

A: To Prove: x is a rational number if and only if 2x-1 is a rational number.

Q: 2. Let x be irrational real number. Prove that 2x+3 is irrational

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Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

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