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: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: How can I prove that if x is a positive irrational number, then 2x-1 is also irrational

A:

Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: If a and b are integers, then a/b is always a rational number. True False

A:

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

A:

Q: Derive one-one correspondence between the real numbers in (0,1) and the set of irrational numbers in…

A: Both the sets that is real number in 0,1 and the set of irrational numbers in 0,1 are uncountable as…

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

A:

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

A:

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

A: This question from number theory.

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

A:

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

A: We need to find the maximum value of a1x1+a2x2+a3x3+...+anxn to the condition that…

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

A:

Q: 2. A pay phone at a railroad station charges $0.55 for the first five minutes (or part of) and $0.25…

A:

Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

A:

Q: 3. A polynomial containing exactly two terms is a(n) .......................

A: We know that, A polynomial containing exactly one term is a monomial. A polynomial containing…

Q: 2. The set of all rational numbers in the interval (0, 1).

A:

Q: 8. Which of the following number lines depicts the solution to - (x- 5)(x- 8) > 0? + R а. 1 3 5 7 9…

A:

Q: (b) Prove that 27 (2⁹⁹ +1).

A:

Q: 1. Find the number of zeroes at the end of 100!

A: Introduction :Here, we have asked for the number of zeros at the end of 100 !

Q: Please don't provide handwritten solution ..

A: The symbol Γ(n) represents the gamma function, which is an extension of the factorial function to…

Q: Find the number of nonnegative integer solutions to x₁ + x₂ + x3 + x4 + x5 ≤ 23. Find the number of…

A: We have to find the number of non negative integer solution of Note: According to the guidelines we…

Q: 2. The sum of three consecutive positive odd integers is always a multiple of (A) 3 (B)5 (C)7 (D)9…

A:

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

A: Given: The given statement ∀x∀yy>x→∃zz is a rational number∧x<z<y To find: we need to…

Q: 9. Solve In (x+5) 2 2. O A. (x | x 2 12.3891} B. (x | x 2 2.3891} O C. (x |xs 12.3891} O D. {x |x s…

A: Given:         ln(x+5)≥2

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: 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: 21 84 = 1, where x and y are positive real numbers, find the smallest possible y of x + y. If + X…

A: Given   21x+84y=1

Q: 3. Find all r so that r-1+r-2| < 5.

A:

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

A: Consider ab,cd are rational numbers with a, b, c, d are integers. Now add these two numbers ab+cd…

Q: For the following claim Below you must disprove it. "For all integers x, x^2 + 1 is odd" Use the…

A: for example take x=3 .  then x^2 +1=10 , which is even.

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&lt;8 Then: b&lt;8⇒-b&gt;-8⇒15-b&gt;15-8⇒a≥15-b&gt;7…

Q: 1. Find L-'[l/(p² + a?)²] by convolution.

A: Let 1p2+a22=1p2+a2p2+a2=1p2+a2p2+a2= FpGp      Then ft=gt=sinata

Q: n(n-1) 2 this statement to justify the following. (n+³) = - :)-( (equation 1). 2 (n + 3)(n + 2) 2…

A:

Q: The sum of three unequal negative integers is - 18. What is the smallest possible value of the…

A: Since the three numbers are negative and unequal , to get the smallest possible value among the…

Q: Prove by contradiction that a real number that is less than every positive real number cannot be…

A:

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

A: