×

# Factors, Divisibility, Remainders

Suppose $$m$$ and $$n$$ are integers such that $$m$$ is divisible by $$8$$ and $$n$$ is divisible by $$5$$. Which of the following integers must be divisible by $$40?$$

$$\begin{array}{r r l} &\mbox{I.} & mn \\ &\mbox{II.} & 5m - 8n \\ &\mbox{III.} & 8m - 5n \\ \end{array}$$

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

If $$n$$ is a positive integer, what is the remainder when $$4n + 11$$ is divided by $$4?$$

(A) $$\ \ 0$$
(B) $$\ \ 1$$
(C) $$\ \ 3$$
(D) $$\ \ 4$$
(E) $$\ \ 11$$

Suppose $$m$$ and $$n$$ are both positive integers strictly greater than 1. If $$m$$ divides both $$n+9$$ and $$n+22$$, which of the following is a possible value of $$m$$?

(A) $$\ \ 1$$
(B) $$\ \ 9$$
(C) $$\ \ 13$$
(D) $$\ \ 22$$
(E) $$\ \$$None of the above

How many positive integers $$k$$ are there such that dividing $$48$$ by $$k$$ leaves a remainder of $$4$$?

(A) $$\$$ One
(B) $$\$$ Two
(C) $$\$$ Three
(D) $$\$$ Four
(E) $$\$$ Five

Let $$n$$ be a positive integer such that the remainder of $$3n + 6$$ upon division by $$4$$ is $$1.$$ Which of the following is a possible value for $$n$$?

(A) $$\ \ 0$$
(B) $$\ \ 2$$
(C) $$\ \ 4$$
(D) $$\ \ 5$$
(E) $$\ \ 6$$

×