Assignment 1

Submitted on 28 August 2024.

Question 1

Review Lesson 1.1.

Question 2

Give a precise statement of The Division Algorithm.

Question 3

Complete the following theorem: Let \( n \in \mathbb{Z} \) with \( n \neq 0 \). Integers \( a \) and \( c \) leave the same remainder when divided by \( n \) if and only if WHAT?

Question 4

Let \( a \in \mathbb{Z} \) be an arbitrary integer. (You don’t get to choose \( a \).) Briefly prove that \( a - 1 \) and \( a + 20 \) leave the same remainder when divided by 7. Which theorem did you use in your proof? Give two numbers between them that leave that same remainder (your answer will have an \( a \) in it).

Question 5

Consider a linear function \( f(x) = mx + b \) with domain \( \mathbb{Z} \). Find \( m \) and \( b \) so that the range of this function is the set of all integers that have remainder 2 when divided by 5.

Hint: The answer is not \( f(x) = 4x + 7 \), for instance, since the range of this function is the set \( f(\mathbb{Z}) = \{4x + 7: x \in \mathbb{Z}\} = \{...,-5, -1, 3, 7, ...\} \), and as you can see, this is not the set of integers with remainder 2 mod 5. There are infinitely many correct (and incorrect) answers to this question.

Question 6

Give a clear, complete, and well-written proof that the square of any integer \( a \) is of the form \( 3k \) or of the form \( 3k + 1 \) for some integer \( k \).

Question 7

Find the quotient and remainder when \( a \) is divided by \( b \). Write the equation from the Division Algorithm that they satisfy, using numbers, not letters.

  • (a) \( a = 8, b = 3 \) — 2 points
  • (b) \( a = -8, b = 3 \) — 2 points

Question 8

What day of the week will it be in 365 days if today is Monday? Explain your answer using the ideas of quotient and remainder.

Question 9

Find \( i^{83} \) (where \( i = \sqrt{-1} \)) and explain your answer using quotients and remainders.

Question 10

Find 4 solutions of the equation \( i^k = i^3 \) (where \( k \) is the variable, an integer). Now go further and describe all the solutions: For which integers \( k \) is \( i^k = i^3 \)? You can use the roster method or set-builder notation to describe your answer.

Question 11

Let’s review some logic. Suppose we have a conditional statement: “If \( p \) then \( q \).”

  1. (a) State the converse.
  2. (b) State the contrapositive.
  3. (c) Which one of these is equivalent to the original statement?

Question 12

How do we negate conditional statements? For your answer, give the negation of the conditional statement, “If \( p \) then \( q \).” (Of course, don’t just write “not” in front of it.)