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If the number of integers \(x\) that satisfy \(\left\lvert 8-\frac{x}{n} \right\rvert<3\) is \(17,\) what is the value of the positive integer \(n?\)

(A)\(\ \ 1\)

(B)\(\ \ 2\)

(C)\(\ \ 3\)

(D)\(\ \ 4\)

(E)\(\ \ 5\)

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If \(a+b=-2\) and \(a^2+b^2=5,\) what is the value of \(a^5+b^5?\)

(A) \(-68\)

(B) \(-55.5\)

(C) \(-54.5\)

(D) \(-24.5\)

(E) \(\ \ 24.5\)

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If \(a-b=-4\) and \(ab=3,\) what is the value of \[\frac{1}{a^3}-\frac{3}{a^2b}+\frac{3}{ab^2}-\frac{1}{b^3}?\]

(A) \(\ \ \frac{40}{27}\)

(B) \(\ \ \frac{64}{27}\)

(C) \(\ \ \frac{112}{27}\)

(D) \(\ \ \frac{64}{9}\)

(E) \(\ \ \frac{112}{9}\)

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If \(x, y\) and \(z\) are positive numbers such that \(x^2=y^5\) and \(y^{-\frac{1}{3}}=z^{-\frac{1}{5}},\) what is \(\sqrt{z^3}\) in terms of \(x?\)

(A) \( \ \ x^{-\frac{1}{2}} \)

(B) \( \ \ x^{\frac{1}{2}} \)

(C) \( \ \ x^{\frac{2}{3}} \)

(D) \( \ \ x \)

(E) \( \ \ x^{\frac{3}{2}} \)

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The following two lines in the \(xy\)-plane are perpendicular to each other: \[\begin{array} &3^{2a}x+27^{-3a+1}y-7=0, &3^{-5a+1}x-9^{a-3-\frac{1}{2a}}y+4=0.\end{array}\] What is the real number \(a?\)

(A) \( \ \ -1 \)

(B) \( \ \ -\frac{1}{2} \)

(C) \( \ \ \frac{1}{2} \)

(D) \(\ \ 1 \)

(E) \(\ \ \frac{3}{2} \)

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