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If both n and $$\frac{72}{n}$$ are positive integers, then how many values could n have?
How many of the integers from 1 to 100, inclusive, are not multiples of 3 or 7?
If n is a positive integer greater than 1, then n($$n^{2}$$-1) must be a multiple of which of the following integers?

Indicate all such numbers.

Quantity A

The remainder when $$2^{32}$$ is divided by 3

Quantity B

1


Quantity A

The remainder when $$7^{22}$$ is divided by 5

Quantity B

4


$$x \gt 0$$

$$y \gt 0$$

Quantity A

($$\sqrt{x}$$)($$\sqrt{y}$$)

Quantity B

$$\sqrt{x+y}$$


$$p$$, $$s$$, and $$t$$ are probabilities and $$0 \lt p \lt s \lt t$$.

Quantity A

$$p+st$$

Quantity B

$$s(p+t)$$


Quantity A

$$\frac{111}{1,111}$$

Quantity B

$$\frac{1,111}{11,111}$$


$$d$$ is a integer greater than $$1$$.

Quantity A

$$(d^{2})!$$

Quantity B

$$(d!)^{2}$$


Which of the following values of $$x$$ satisfies the equation $$\frac{x}{2}=n!$$ for some positive integer $$n$$?


The boxplot above summarizes a list of 240 numbers. Which of the following statements must be true?

Indicate all such expressions.
The range of a list of 13 numbers is 110. The range of 10 of these numbers is 80.

Quantity A

The range of the remaining 3 numbers

Quantity B

30


Set A={22, 23, 24, 25, 26, 27}
Set B={222, 223, 224, 225, 226, 227}

Quantity A

The standard deviation of integers in Set A

Quantity B

The standard deviation of integers in Set B


Sample P: 0.8, 0.9, 0.9, 1.0

Sample Q: 1.0, 1.0, 1.0, 1.0

Sample R: 0.8, 0.8, 1.0, 1.0

Sample S: 0.7, 0.7, 0.9, 0.9

Sample T: 0.9, 1.0, 1.0, 1.1

Five samples of 4 washers each were taken from a production line. The inside diameter of each washer was recorded to the nearest 0.1 centimeter, as shown above. For which of the samples is the standard deviation of the measurements greater than the standard deviation of the measurements for sample P?

Indicate all such samples.

Quantity A

(40!)(30!)

Quantity B

(60!)(20!)


Three coins—two $$10$$-cent coins and one $$5$$-cent coin—are to be flipped simultaneously. For each of the three coins, the probability that the coin will land heads up is $$\frac{1}{2}$$. What is the probability that the total value of the coins that will land heads up is $$15$$ cents?
Give your answer as a fraction.


The circle in the xy-plane shown has radius 5 and center O. Line l has a slope of $$\frac{1}{2}$$ and passes through the center of the circle. P is the point in the first quadrant at which l intersects the circle. Which of the following represents the coordinates of point P?
If x=2y+1, and y=2w, where w, x, and y are integers, which of the following must be an odd integer?

Quantity A

$$(24-25)^{50}$$

Quantity B

$$(77-76)^{-1}$$


Let $$\frac{1}{3}$$+$$\frac{1}{5}$$+$$\frac{1}{7}$$+$$\frac{1}{9}$$+$$\frac{1}{11}$$=$$\frac{m}{(3)(5)(7)(9)(11)}$$, where m is a positive integer. What is the remainder when m is divided by 5?

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