A whole number divisible by 24 is also divisible by any factor of 24, including 8.

A whole number is divisible by 8 if and only if its last three digits form a number which itself is divisible by 8. We can try each of the three digits: 

None of the choices yields a number divisible by 8. Since 8 cannot be a factor of this number,  neither can 24.

A number divisible by 36 must be divisible by any factor of 36; two such factors are 4 and 9, so we can test for divisibliity by both.

The number is divisible by 4 if and only if its last two digits form a number divisible by 4. 32, 52, 72, and 92 are all divisible by 4:

Therefore, all of the choices make the number divisible by 4. Therefore, we see which one makes the number divisible by 9, which happens if and only if the sum of the digits is a multiple of 9. If we let be the missing digit, then the digit sum is

.

We can test each digit by substituting for :

Since 18 is the only digit sum divisible by 9, 3 is the correct digit to fill in the box.

 is the complement of , making  the set of all persons not in  - that is, the set of all people who do not own a canoe.  is the set of all persons who own a motorboat. To be in , the union of the sets, a person must be in one or the other set, or both, or, equivalently, the person must either not own a canoe or must own a motorboat, or both. Donald is in  by virtue of owning a motorboat. Ed, however, owns a canoe, but does not own a motorboat, so he does not meet either qualification. 

First, the table at which Mr. Gregory is seated must be decided - there are two alternatives. Mrs. Gregory will be at the other table.

Then it must be decided which of the four seats Mr. Gregory will occupy at his table, and which of the four seats Mrs. Gregory will occupy at hers.

Finally, it must be decided how the six children will be arranged in the remaining six seats - this can happen in

ways.

Apply the multiplication principle; there will be 

possible arrangements meeting the specifications.

In the range of 1 to 10, the composite numbers - the numbers with at least one factor other than 1 and itself - are 4, 6, 8, 9, and 10. Therefore, there are five composite numbers represented by two balls each, and five noncomposite numbers (1, 2, 3, 5, 7) represnted by 1 ball each. The total number of balls is

Among the odd numbers, 1, 3, 5, and 7 are represented by one ball each; 9 is represented by two, for a total of 6 balls out of 15. This makes the probability of drawing an odd number

The following choices need to be made:

First, the decision must be made as to the table at which the Johnsons will sit. There are two choices.

Next, the decision must be made as to which seats the Johnsons will occupy at that table. There are two choices - the "north-and-south" seats and the "east-and-west" seats.

Third, the decision must be made which one will sit where; there are two choices now.

Fourth, the decision must be made as to how the six children will be seated - this will be

.

By the multiplication principle, the number of seating arrangements with the Johnsons sitting across from each other will be

An integer is divisible by 11 if and only if its alternating digit sum is divisible by 11. If we let  be the missing digit, then the alternating digit sum is 

Through trial and error, we see that no value of  from 0 to 10 makes this a multple of 11. This makes none the correct choice.

Let , , , and  represent the set of students who like "The Walking Dead", "Boardwalk Empire", "The Sopranos", and "Deadwood", repsectively.

Everyone who likes "The Walking Dead" likes "Boardwalk Empire", so .

Everyone who dislikes "The Sopranos" likes "Deadwood", so .

Nobody likes both "Boardwalk Empire" and "The Sopranos", so  and  have no elements in common - they are disoint sets. One way of saying this is that everyone who likes "Boardwalk Empire" dislikes "The Sopranos" - that is, . 

These statements can be put together as follows:

We can represent Philip as . Philip likes "The Walking Dead", so it follows that

, which means he does not like "The Sopranos",

and

, which means he likes "Deadwood".

Let , , , and  represent students who like Diddy, Prince, Weezer, and Bruce Springsteen, respectively.

Every student who likes Diddy also likes Prince, so .

Every student who likes Weezer also likes Bruce Springsteen, so 

No student likes both Prince and Bruce Springsteen, so  and  have no elements in common - they are disjoint sets.

A Venn diagram for this scenario is below:

Note that since Jerry ("j") is an element of , he cannot be an element of  or . Therefore, he does not like Diddy or Prince.

The number of miles per hour over the limit is the difference of the number of miles per hour the driver was going and the speed limit - this is . The fine is $75 plus $4 times this difference: