[
  {
    "kind": "Proposition",
    "label": "1.6.1",
    "first_line": "Proposition 1.6.1."
  },
  {
    "kind": "Proposition",
    "label": "1.6.2",
    "first_line": "Proposition 1.6.2. Let a, b, c, u, and v denote integers."
  },
  {
    "kind": "Definition",
    "label": "1.6.3",
    "first_line": "Definition 1.6.3. A natural number is prime if it is greater than 1 and not"
  },
  {
    "kind": "Proposition",
    "label": "1.6.4",
    "first_line": "Proposition 1.6.4. Any natural number other than 1 can be written as a"
  },
  {
    "kind": "Theorem",
    "label": "1.6.6",
    "first_line": "Theorem 1.6.6. There are infinitely many prime numbers."
  },
  {
    "kind": "Proposition",
    "label": "1.6.7",
    "first_line": "Proposition 1.6.7. Given integers a and d , with d \u0015 1, there exist unique"
  },
  {
    "kind": "Definition",
    "label": "1.6.8",
    "first_line": "Definition 1.6.8. A natural number d is the greatest common divisor of"
  },
  {
    "kind": "Proposition",
    "label": "1.6.9",
    "first_line": "Proposition 1.6.9. For integers n and m, let"
  },
  {
    "kind": "Lemma",
    "label": "1.6.10",
    "first_line": "Lemma 1.6.10. Let m and n be nonzero integers. If a natural number d"
  },
  {
    "kind": "Proposition",
    "label": "1.6.11",
    "first_line": "Proposition 1.6.11. The natural number nr is the greatest common divisor"
  },
  {
    "kind": "Corollary",
    "label": "1.6.13",
    "first_line": "Corollary 1.6.13. Let m and n be nonzero integers, and write d D"
  },
  {
    "kind": "Definition",
    "label": "1.6.14",
    "first_line": "Definition 1.6.14. Nonzero integers m and n are relatively prime if"
  },
  {
    "kind": "Corollary",
    "label": "1.6.15",
    "first_line": "Corollary 1.6.15. Two nonzero integers m and n are relatively prime if"
  },
  {
    "kind": "Corollary",
    "label": "1.6.17",
    "first_line": "Corollary 1.6.17. Suppose that a and b are relatively prime natural num-"
  },
  {
    "kind": "Proposition",
    "label": "1.6.18",
    "first_line": "Proposition 1.6.18. If p is a prime number and a is any nonzero integer,"
  },
  {
    "kind": "Proposition",
    "label": "1.6.19",
    "first_line": "Proposition 1.6.19. Let p be a prime number, and a and b nonzero inte-"
  },
  {
    "kind": "Corollary",
    "label": "1.6.20",
    "first_line": "Corollary 1.6.20. Suppose that a prime number p divides a product"
  },
  {
    "kind": "Theorem",
    "label": "1.6.21",
    "first_line": "Theorem 1.6.21. The prime factorization of a natural number is unique."
  },
  {
    "kind": "Definition",
    "label": "1.6.22",
    "first_line": "Definition 1.6.22. A natural number d is the greatest common divisor of"
  },
  {
    "kind": "Lemma",
    "label": "1.6.23",
    "first_line": "Lemma 1.6.23. Given nonzero integers a1 ; a2 ; : : : ; an (n \u0015 2), there is a"
  },
  {
    "kind": "Proposition",
    "label": "1.6.24",
    "first_line": "Proposition 1.6.24. The greatest common divisor of nonzero inte-"
  },
  {
    "kind": "Definition",
    "label": "1.6.25",
    "first_line": "Definition 1.6.25. We say that non-zero integers a1 ; : : : ; an are relatively"
  },
  {
    "kind": "Lemma",
    "label": "1.6.26",
    "first_line": "Lemma 1.6.26. Let a1 ; : : : ; an be pairwise relatively prime integers, and"
  }
]