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Basic Mathematics

# Checking Cases

Which of the following could be an internal angle in a regular polygon?

$$\begin{array}{r r l} &\text{I.}\ & 108 ^ \circ\\ & \text{II.}\ & 120 ^ \circ\\ &\text{III.} &167 ^ \circ\\ \end{array}$$

(A)$$\ \$$ I only
(B)$$\ \$$ III only
(C)$$\ \$$ I and II only
(D)$$\ \$$ II and III only
(E)$$\ \$$ I, II, and III

If $$n$$ is a perfect square, which of the following could be true?

$$\begin{array}{r r l} &\text{I.}\ & n\ \text{is a perfect cube.}\\ & \text{II.}\ & n^2 + 6\ \text{is a perfect square.}\\ & \text{III.}\ & n^2 + 1\ \text{is a multiple of 3.}\\ \end{array}$$

(A)$$\ \$$ I only
(B)$$\ \$$ II only
(C)$$\ \$$ I and II only
(D)$$\ \$$ I and III only
(E)$$\ \$$ I, II, and III

If $$x^3 < 8 x + 9$$, which of the following could be a value of $$x$$?

$$\begin{array}{r r l} & \text{I.} & 8\\ & \text{II.} & 0\\ & \text{III.} & -8\\ \end{array}$$

(A)$$\ \$$ I only
(B)$$\ \$$ III only
(C)$$\ \$$ I and II only
(D)$$\ \$$ I and III only
(E)$$\ \$$ II and III only

For all numbers $$a$$ and $$b$$, let $$a \heartsuit b$$ be defined by $$a \heartsuit b = ab + a -b$$. For all numbers $$x$$ and $$y$$, which of the following must be true?

$$\begin{array}{r r l} &\text{I.}\ & x \heartsuit y = y \heartsuit x\\ &\text{II.}\ & x \heartsuit y = (-y) \heartsuit (-x)\\ &\text{III.}\ & x \heartsuit y = (y+3) \heartsuit (x-3)\\ \end{array}$$

(A)$$\ \$$ II only
(B)$$\ \$$ III only
(C)$$\ \$$ II and III only
(D)$$\ \$$ I, II, and III
(E)$$\ \$$ None of the statements

If $$P$$ is a prime number greater than 10, which of the following could represent another prime number for some value of $$P$$?

$$\begin{array}{r r l} &\text{I.}\ &P+26\\ &\text{II.}\ & 5P\\ &\text{III.}\ & P^2 + 1\\ \end{array}$$

(A)$$\ \$$ I only
(B)$$\ \$$ III only
(C)$$\ \$$ I and II only
(D)$$\ \$$ I and III only
(E)$$\ \$$ II and III only

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