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Numbers and Operations

SAT Numbers Perfect Score

\(P(n)\) is a function defined as follows: \[P(n)=\text{(Sum of all prime factors of }n\text{)},\] where \(n\) is a positive integer. What is the value of the following expression: \[ P(36)+P(225)+P(289)? \]

(A)\(\ \ \) \(27\)
(B)\(\ \ \) \(30\)
(C)\(\ \ \) \(33\)
(D)\(\ \ \) \(38\)
(E)\(\ \ \) \(801\)

If \(a\) and \(b\) are integers, what is the number of ordered pairs \((a,\ b)\) that satisfy the following statements:

\(\begin{array}{r r l} &\text{I.}&1<3^b<2^a\\ &\text{II.}&a^2<b^3<65?\\ \end{array}\)

(A)\(\ \ \) \(2\)
(B)\(\ \ \) \(3\)
(C)\(\ \ \) \(4\)
(D)\(\ \ \) \(10\)
(E)\(\ \ \) \(13\)

Functions \(O(n)\) and \(E(n)\) are defined as follows: \[\begin{align}O(n) &= 2n -1\\E(n)&= 2n,\end{align}\] where \(n\) is an integer. Which of the following statements must be true?

\(\begin{array}{r r l} &\text{I.} &O(3)=5.\\
&\text{II.} &\text{There always exists an integer }p \text{ that satisfies } \\ & & \frac{E(3q-1)}{2} =O(p) \text{ for any even number }q.\\
&\text{III.} & \text{There always exists an integer }p \text{ that satisfies }\\&&E(q)O(q)=E(p)\text{ for any integer }q.\\
\end{array}\)

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

What is the value of the following series: \[\left\lfloor \frac{O(1)}{3} \right\rfloor+\left\lfloor \frac{O(2)}{3} \right\rfloor+\left\lfloor \frac{O(3)}{3} \right\rfloor+\cdots+\left\lfloor \frac{O(81)}{3} \right\rfloor,\] where \(O(n)=2n-1\) and \(\lfloor \text{ } \rfloor\) is the floor function?

(A)\(\ \ \) \(756\)
(B)\(\ \ \) \(2160\)
(C)\(\ \ \) \(2187\)
(D)\(\ \ \) \(2322\)
(E)\(\ \ \) \(6561\)

If the operation \(\star\) is defined as \(m\star n = m^2n\) for all integers \(m\) and \(n,\) which of the following statements must be true?

I. There exists only one integer \(x\) satisfying \(x\star a=a,\) where \(a\) is a positive integer.
II. For any integers \(b, c\) and \(d,\) the following always holds: \[b\star(c\star d)=(b\star c)\star d.\] III. Given \(f_0(x)=x\) and \(f_n(x)=f_{n-1}(x)\star 2,\) \[f_0(2)\cdot f_1(2)\cdot f_2(2) \cdots f_{n-1}(2)=2^{2^{n+1}-n-2}.\]

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

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