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In a list of $$16$$ consecutive multiples of $$5$$, ordered from least to greatest, the difference the $$2$$nd and $$15$$th numbers is what fraction of the difference between the first and last numbers?
Sixteen fence posts are equally spaced in a straight line along a property line. The distance between the first post and the last is $$240$$ feet.

Quantity A

The distance between the $$5$$th post and the $$8$$th post

Quantity B

$$45$$ feet


There are $$16$$ integers between $$1$$ and $$200$$ that are multiples of $$12$$. If one of these integers is randomly chosen, what is the probability that the chosen integer will be a multiple of $$9$$?
How many multiples of $$5$$ are there from $$55$$ to $$505$$, inclusive?

Quantity A

The number of multiples of $$3$$ between $$1$$ and $$10,000$$

Quantity B

The number of multiples of $$7$$ between $$1$$ and $$23,000$$


How many positive two-digit integers are not divisible by $$4$$?
Fred receives $$40$$ channels, numbered with the integers from $$1$$ to $$40$$, on his television set. If he is to select a channel randomly, what is the probability that the channel number will be divisible by $$4$$ or $$5$$ or both?
$$S$$ is the set of all 2-digit integers that are multiples of $$3$$.

Quantity A

The number of even integers in $$S$$

Quantity B

$$15$$


Quantity A

The number of positive integers that are factors of both $$24$$ and $$30$$ but are not factors of $$20$$

Quantity B

The number of positive integers that are factors of both $$20$$ and $$30$$ but are not factors of $$24$$


If n is an integer and $$\frac{n-5}{2}$$is an integer, which of the following must be an integer?
$$h$$ and $$m$$ are positive integers.
$$(10^{50})h+m$$ is divisible by $$9$$.

Quantity A

The remainder when $$h+m$$ is divided by $$9$$.

Quantity B

$$5$$


What is the least positive integer $$k$$ such that $$\frac{(4)(5)(6)(7)(8)(9)+2k}{10}$$ is an integer?

Quantity A

The remainder when the sum of $$3$$ consecutive positive integers is divided by $$2$$

Quantity B

The remainder when the product of $$3$$ consecutive positive integers is divided by $$2$$


If $$w$$, $$x$$, $$y$$, and $$z$$ are consecutive positive integers, where $$w \lt x \lt y \lt z$$, which of the following statements must be true?

Indicate all such statements.
The sum of an integer $$k$$ and the square of $$k$$ CANNOT be a

Quantity A

The smallest positive integer that is a multiple of both $$3$$ and $$12$$

Quantity B

The smallest positive integer that is a multiple of both $$3$$ and $$4$$


$$n$$ is a multiple of $$6, 7$$, and $$9$$.
$$200 \leq n \leq 300$$

Quantity A

$$n$$

Quantity B

$$250$$


Quantity A

The least common multiple of $$10, 15$$, and $$16$$

Quantity B

The least common multiple of $$20, 15$$, and $$16$$


$$E$$ is the set of all positive common multiples of $$7$$ and $$11$$ that are less than $$1,900$$.

Quantity A

The number of integers in $$E$$ that are divisible by $$6$$

Quantity B

The number of integers in $$E$$ that are divisible by $$5$$


$$n$$ is a positive integer.
The remainder when $$5n + 2$$ is divided by $$3$$ is $$1$$.
The remainder when $$5n + 1$$ is divided by $$4$$ is $$1$$.

Quantity A

$$n$$

Quantity B

$$3$$


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