. Let a be a positive number. Prove that for each real number x there is an integer n such that na < x < (n + 1)a.

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Prove For every real number m, there exists an integer a such that |m−a|<1.

A: If x is any non integral real number then [x] is the largest integer less than x and…

Q: Let S be a linear transformation from R³ to R² with associated matrix Let T be a linear…

A: Let  be a linear transformation from  to  with associated matrix .Let  be a linear transformation…

Q: Let x be an integer. Prove that 2|x and 3|x if and only if 6|x. Generalize and prove.

A: According to the given information, let x be an integer. It is required to prove that:

Q: Noah invests $3,600 for two years.   The account pays 0.43% monthly compound interest. To the…

A: Given the given invest=$3,600 Interest rate=0.43%(compounded monthly) Time=2 years Apply formula…

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

A: Let m be any integer and x be any real number that is not an integer.

Q: Let 7 be a real number. Which of the following is equal to T T -2 (ED)*+E)* r2 3 0 None of the these…

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Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

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Q: Let 0 < a < 1 be a real number. Define a₁ = 1+a, Prove that Vn € N, 1 ≤n, an+1 = 1<an< 1 - an + a, -…

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Q: Let a,b and x be real numbers with a0. Show that there exists a rational number t such that a <

A: We know that the real numbers are densely ordered. This means that between any two distinct real…

Q: Let x1, x2,...,n be real numbers. Prove that |X1 + X2 + ··· + Xn] ≤ |X1| + |X2| + · · · + |xn|·

A: The objective of this question is to prove the triangle inequality for a sequence of real numbers.…

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: Z/mZz Zm

A: We need to prove that Z / mZ is isomorphic to Zm .

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n &gt; 0.

A: Let a be a non zero real number , where m and n are integers…

Q: () is even. 3. Prove that for all positive integers n, the number

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Q: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

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: 5 A clerk sold three pieces of one type of ribbon to different customers. One piece was 3 yards…

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Q: 1

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Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: Prove that the roots of X2+(1-K)x+k-3=0 are real for all values of k?

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Q: 6: Prove by contradiction that for any positive integer x, x+ 49 x ≥ 14.

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Q: Let A and B be sets of real numbers and let B be finite. Prove that if x is an accumulation point of…

A: Let A and B be sets of real numbers and let B be finite. Prove that if x is an accumulation point of…

Q: Let n be an integer. Apply any methods to prove that n? can be written as either 3k or 3k + 1 for…

A: By the quotient remainder theorem, any integer n can be written as n= 3q+r, that is n=3q or n= 3q+1…

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: Prove that there are integers x and y such that 10x-13y=1

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Q: Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For…

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Q: For every real number there exist an integer m such that m<b ylgn ihi

A: Use Archimedean property of real numbers.

Q: If a and b are positive integers such that gcd(a,b)=13, then find the sum of all possible distinct…

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Q: Let A be the subset of the real numbers given by A = {1 + a : a ∈ R and a > 0}. Show that inf A =…

A: A={1+a:a∈R and a&gt;0}Now,1+a&gt;1 for all a∈R and a&gt;0Thus 1 is a lower bound of A.We shall show…