Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is $\underline{\hspace{2cm}}$.

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is $\underline{\hspace{2cm}}$.

Let $n$ be a two-digit natural number such that the remainder of $n ^ { 3 }$ divided by 66 is $n$. We are to find the number of such $n$ 's and to find the prime numbers among them.
From the conditions we have
$$n ^ { 3 } = \mathbf { A B } p + n \quad ( 0 < n \leqq \mathbf { C D } ) ,$$
where $p$ is the integer quotient of $n ^ { 3 }$ divided by 66 . This can be transformed into
$$n ( n - 1 ) ( n + 1 ) = \mathrm { AB } p$$
Since either $n - 1$ or $n$ has to be a multiple of $\mathbf { E }$ and either $n - 1 , n$ or $n + 1$ has to be a multiple of $\mathbf { F }$, and furthermore $\mathbf { E }$ and $\mathbf { F }$ are mutually prime, we know that $n ( n - 1 ) ( n + 1 )$ is a multiple of $\mathbf { G }$. (Write the answers in the order $1 < \square < \mathbf { E } < \mathbf { F } < \mathbf { G }$.) Hence one of $n - 1 , n$ and $n + 1$ must be a multiple of $\mathbf{HI}$.
So, since $n \leqq \mathrm { CD }$, the number of $n$ 's where $n - 1$ is a multiple of $\mathbf{HI}$ is $\mathbf{J}$, where $n$ is a multiple of $\mathbf{HI}$ is $\mathbf { K }$, and where $n + 1$ is a multiple of $\mathbf{HI}$ is $\mathbf { L }$.
Thus, the number of $n$ 's is $\mathbf { M N }$ and the prime numbers among them are $\mathbf { O P } , \mathbf { Q R }$, $\mathbf{ST}$, in ascending order.

$$\begin{aligned} & A = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 3 \right\} \\ & B = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 5 \right\} \end{aligned}$$
The sets are given. Accordingly, how many elements does the difference set $\mathrm { A } \backslash \mathrm { B }$ have?
A) 33
B) 32
C) 30
D) 28
E) 27

In the following table consisting of 100 unit squares numbered from 1 to 100, some squares will be painted.

1	2	3	$\ldots$	10
11	12	13	$\ldots$	20
.	.	.		.
.	.	.		.
.	.	.		.
91	92	93	..	100

Squares with even numbers are painted yellow, squares that are multiples of 3 are painted red, and squares that are multiples of 5 are painted blue.
For a square to be orange, it must be painted only yellow and red.
Accordingly, how many unit squares in the table are orange?
A) 8
B) 12
C) 13
D) 15
E) 18

When 6 of the integers from 1 to 9 are placed in the boxes below such that each box contains a different number, all equalities are satisfied.
$$\begin{aligned} & \square + \square = 5 \\ & \square - \square = 5 \\ & \square : \square = 5 \end{aligned}$$
Accordingly, what is the sum of the unused integers?
A) 23
B) 21
C) 19
D) 17
E) 15

A three-digit natural number whose digits are different from each other and from zero is called a middle-divisible number if the digit in the tens place divides the digits in the other places. For example, 428 is a middle-divisible number. Accordingly, what is the difference between the largest middle-divisible number and the smallest middle-divisible number?
A) 723
B) 727
C) 736
D) 742
E) 745