If α and β are two real numbers, we say that β is equivalent to α if there are integers a, b, c, and d such that ad - bc = ±1 and β = (aα+b)/(cα+d) Show that a real number α is equivalent to itself.

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

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Q: 2. Find an explicit formula for аn 3 За,-1+ 10ар-2 and ao %3D 0, ај 3D 7 An where

A: The solution is given as

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Q: : 2 < x < 4} and B = {x € R : –4 < x < 1}. Prove

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Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

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Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

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Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

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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: Given: 5(x + 2) = -3x - 6 Prove: x = -2

A: 7

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Q: If x is any real number, then cosh x Σ n=0 n! + 198 Σ n=0 (-x)" n!

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

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Q: For which values of a does ax = 1 have a solution in Z5?

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Q: 1. Let x, y, and z be real numbers. Prove the following: [a] -(-x)=x [b] (-x)y = -(xy) and (-xX- y)=…

A: "By Bartleby rules, only first three sub-parts of question 1 have been answered"

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A: Find the value

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Q: integer of the form okts is also but not' conversely

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Q: If a, b, c are non-zero real numbers, then bc ca ab ca ab bc vanishes, when ab bc ca ca 1.1 -+- 0…

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Q: 1 If x is a real number, then is x² + x4 equal to а. V1 + x² b. 1 xV1+ x? c. xV1+x² d. æ|/1+ x² 1 е.…

A: Let's find.

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Q: Q2 1-The Euler's number (e) is a * rational number a. True b. False

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Q: 2. How many positive integer solutions are there to Xị + x2 + x3+ x4 < 100

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Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

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Q: 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…

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Q: 7. For all odd integers x and y, x² +3y = 0 (mod8) . 8. The equation x* +x² +1=0 has no real…

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Q: 5.46. Prove that there exists a unique real number solution to the equation x O between a 2/3 and r…

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Q: For every real number x+1, n-1 1+x+x² +x' +. x´+x' +...+x" for all neN. 1-x

A: The question has a mistake , as in the R.H.S the power of x must be (n+1).

Q: Given: 5(2x – 8) = 30; Prove: x = 7 -

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Q: 3 and for n > 2, an = 2an-1+5, express an in terms of n. 1. Let, a1

A: The solution is given as

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Q: Prove that ifx^2−6x+ 3is odd, then x is even

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Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation

Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

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Q: How many non-negative integer solutions are there for the following equation such that 0 < x1 < 10?…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.