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    "<a target=\"_blank\" href=\"https://colab.research.google.com/github/rzmk/ladderz/blob/main/notebooks/pre-algebra/unit1.ipynb\">\n",
    "  <img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/>\n",
    "</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Unit 1: Factors and multiples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Understanding Factor Pairs\n",
    "\n",
    "> [Link to lesson](https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-factors-mult/v/understanding-factor-pairs).\n",
    "\n",
    "**Write a program that finds all the factor pairs for a number $n$.**\n",
    "\n",
    "- Do not repeat any pairs (e.g., consider `(2, 8)` and `(8, 2)` as the same).\n",
    "- Assume that $n$ is a positive integer greater than or equal to 1 ($n \\in \\mathbb{Z}^+$).\n",
    "\n",
    "For example for $n = 16$, the output of `get_factor_pairs(16)` may be:\n",
    "\n",
    "```\n",
    "{(1, 16), (2, 8), (4, 4)}\n",
    "```\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_factor_pairs(n: int) -> set:\n",
    "    factor_pairs: set = set()\n",
    "    for i in range(1, n + 1):\n",
    "        dividend: int = n\n",
    "        divisor: int = i\n",
    "        quotient: int = int(dividend / divisor)\n",
    "        remainder: int = dividend % divisor\n",
    "        if remainder == 0 and (quotient, divisor) not in factor_pairs:\n",
    "            factor_pairs.add((divisor, quotient))\n",
    "    return factor_pairs\n",
    "\n",
    "assert get_factor_pairs(1) == {(1, 1)}\n",
    "assert get_factor_pairs(15) == {(1, 15), (3, 5)}\n",
    "assert get_factor_pairs(16) == {(1, 16), (2, 8), (4, 4)}\n",
    "assert get_factor_pairs(21) == {(1, 21), (3, 7)}\n",
    "assert get_factor_pairs(26) == {(1, 26), (2, 13)}\n",
    "assert get_factor_pairs(30) == {(1, 30), (2, 15), (3, 10), (5, 6)}\n",
    "assert get_factor_pairs(40) == {(1, 40), (2, 20), (4, 10), (5, 8)}\n",
    "assert get_factor_pairs(42) == {(1, 42), (2, 21), (3, 14), (6, 7)}\n",
    "assert get_factor_pairs(66) == {(1, 66), (2, 33), (3, 22), (6, 11)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding factors of a number\n",
    "\n",
    "> [Link to lesson](https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-factors-mult/v/finding-factors-of-a-number).\n",
    "\n",
    "**Write a program that finds all the factors of a number $n$.**\n",
    "\n",
    "- Assume that $n$ is a positive integer greater than or equal to 1 ($n \\in \\mathbb{Z}^+$).\n",
    "\n",
    "For example for $n = 16$, the output of `get_factors(16)` may be:\n",
    "\n",
    "```\n",
    "{1, 2, 4, 8, 16}\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_factors(n: int) -> set:\n",
    "    factors: set = set()\n",
    "    for i in range(1, n + 1):\n",
    "        dividend: int = n\n",
    "        divisor: int = i\n",
    "        quotient: int = int(n / i)\n",
    "        remainder: int = n % i\n",
    "        if remainder == 0:\n",
    "            factors.add(quotient)\n",
    "    return factors\n",
    "\n",
    "assert get_factors(16) == {1, 2, 4, 8, 16}\n",
    "assert get_factors(120) == {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding factors and multiples\n",
    "\n",
    "> [Link to lesson](https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-factors-mult/v/finding-factors-and-multiples)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Write a program that checks if a number $x$ is a factor of $y$.**\n",
    "\n",
    "- Assume that $x$ and $y$ are positive integers greater than or equal to 1 ($x, y \\in \\mathbb{Z}^+$).\n",
    "\n",
    "For example, for $x = 2$ and $y = 10$, the output of `is_factor(2, 10)` may be:\n",
    "\n",
    "```\n",
    "True\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_factor(x: int, y: int) -> bool:\n",
    "    return y % x == 0\n",
    "\n",
    "assert is_factor(2, 10) == True\n",
    "assert is_factor(3, 10) == False"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Write a program that checks if a number $x$ is a multiple of $y$.**\n",
    "\n",
    "- Assume that $x$ and $y$ are positive integers greater than or equal to 1 ($x, y \\in \\mathbb{Z}^+$).\n",
    "\n",
    "For example, for $x = 20$ and $y = 3$, the output of `is_multiple(20, 3)` may be:\n",
    "\n",
    "```\n",
    "True\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_multiple(x: int, y: int) -> bool:\n",
    "    return x % y == 0\n",
    "\n",
    "assert is_multiple(10, 2) == True\n",
    "assert is_multiple(10, 3) == False"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Write a program that finds all multiples of a given number $n$ up to a given limit $l$.**\n",
    "\n",
    "- Assume that $n$ and $l$ are positive integers greater than or equal to 1 ($n, l \\in \\mathbb{Z}^+$).\n",
    "\n",
    "For example for $n = 3$ and $l = 20$, the output of `get_multiples(3, 20)` may be:\n",
    "\n",
    "```\n",
    "{3, 6, 9, 12, 15, 18}\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_multiples(n: int, l: int) -> set:\n",
    "    multiples = set()\n",
    "    for num in range(n, l + 1):\n",
    "        if num % n == 0:\n",
    "            multiples.add(num)\n",
    "    return multiples\n",
    "\n",
    "assert get_multiples(3, 20) == {3, 6, 9, 12, 15, 18}\n",
    "assert get_multiples(5, 50) == {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}"
   ]
  }
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