#set page(margin: (x: .5in, y: .5in))
#let solve(solution) = [
  #let solution = align(
    center,
    block(
      inset: 5pt,
      stroke: blue + .3pt,
      fill: rgb(0, 149, 255, 15%),
      radius: 4pt,
    )[#align(left)[#solution]],
  )
  #solution
]
#align(center)[
  = CS 3333 Mathematical Foundations
  Homework 1 (100 points)\
  #underline[Price Hiller] *|* #underline[zfp106]
]

*Submission:*
+ Submit a single PDF (not Word) file through Canvas.
+ You can either edit the Word file directly or write down your solution in any other text document using other editors.
+ Convert the Word document or the text document into a single PDF file.
+ If there is any plagiarism, you will lose all points on the questions at first time. In next, you will lose all points in the whole homework.
#line(length: 100%, stroke: .25pt)

*Questions:*
1. Does 41 divide each of these numbers (just answer yes or no)? (16 points)
#solve[#align(center)[]#grid(
    columns: (100pt, 100pt),
    rows: (10pt, auto),
    gutter: 3pt,
    [a) 123 *(yes)*], [b) 92 *(no)*],
    [c) 413 *(no)*], [d) 1640 *(yes)*],
  )]

2. Prove/Show that if $a | b$ and $b | a$, where $a$ and $b$ are integers, then $a = b$ or $a = -b$. (20 points)
  #solve[
    + $b = a k$, where $k$ is an integer
    + $a = b q$, where $q$ is an integer
    + Use $a = b q$
    + Substitute $b$: $a = a k q$
    + Algebra: $k q = 1$
    + ∵ $k$ & $q$ are integers, $k$ and $q$ must be $1$ or $-1$
    + ∴ $a = b$ or $a = -b$
  ]

3. Prove that if $a$, $b$, $c$, and $d$ are integers, where $a ≠ 0$ and $b ≠ 0$, such that $a | c$ and $b | d$, then $a b | c d$. (20 points)
  #solve[
    + $c = k a$, where $k$ is an integer
    + $d = q b$, where $q$ is an integer
    + $c d = (k a)(q b) = (k q)(a b)$
    + $a b | (k q)(a b)$ is true
    + ∴ $a b | c d$
  ]

4. When $x$ is an integer, prove that $1 + x^2 <= (1 + x)^2$ is true or disprove it by providing a counterexample. (20 points)
  #solve[
    + False. Counterexample: let $x = -1$
    + Substitute $x$: $1 + (-1)^2 <= (1 + (-1))^2$
    + Solve: $2 <= 0$, not true
    + ∴ when $x$ is an integer $1 + x^2 <= (1 + x)^2$ is false
  ]

5. Prove that if $a$ is an integer that is not divisible by $3$, denoted as $3 ∤ a$, then $(a + 1)(a + 2)$ is divisible by $3$, denoted as $3 | (a + 1)(a + 2)$. (24 points)
  #solve[
    + Prove: $3 | (a + 1)(a + 2)$
    + Assume $a = 3k + 1$, where $k$ is an integer
    + Substitute $a$: $((3k + 1) + 1)((3k + 1) + 2)$
    + Algebra: $9k^2 + 15k + 6$
    + Factor $3$: $3(3k^2 + 5k + 2)$
    + $3$ times some integer is necessarily divisible by $3$ according to divisibility
    + ∴ If $3 ∤ a$, then $3 | (a + 1)(a + 2)$
  ]