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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

$$3$$


A string of decorative lights consists of red, green and blue bulbs. The red bulbs flash every $$4$$ seconds, the green bulbs flash every $$6$$ seconds, and the blue bulbs flash every $$9$$ seconds. If all the lights flash simultaneously when the string is first lit, how many seconds will it take until all the lights flash simultaneously the next time?

Quantity A

The greatest common factor of $$5$$ and $$16$$

Quantity B

The greatest common factor of $$25$$ and $$256$$


Let $$n$$ be a positive integer such that the greatest common factor of $$n$$ and $$24$$ is $$12$$. Which of the following could be the least common multiple of $$n$$ and $$24$$?
If the sum of $$x$$ and $$5$$ is divided by $$y$$, the result is $$8$$.

Quantity A

$$x$$

Quantity B

$$y$$


When $$10$$ is divided by the positive integer $$k$$, the remainder is $$k$$-2. Which of the following could be the integer $$k$$?
Indicate all such integers.
When the positive integer $$n$$ is divided by $$35$$, the remainder is $$27$$. What is the remainder when $$n$$ is divided by $$7$$?
When the positive integer $$n$$ is divided by $$35$$, the remainder is $$29$$. What is the remainder when $$n$$ is divided by $$7$$?
If the positive integer $$n$$ is divided by $$30$$, the remainder is $$5$$. Which of the following must be divisible by $$30$$?

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