{"id": "goodman-1-6-ex-4", "source_label": "Exercise 1.6.4", "kind": "exercise", "statement": "gcd 60 8 = 4 ∧ 4 = 60 * a + 8 * b ∧ gcd 32242 42 = 2 ∧ 2 = 32242 * c + 42 * d", "prelude": "import Mathlib.Data.Int.GCD", "natural_language": "Compute the greatest common divisor of (a) 60 and 8, and (b) 32242 and 42, and express each as an integer linear combination of the given numbers.", "requires": ["Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": false, "exclude_reason": "Specific numeric computation"}
{"id": "goodman-1-6-ex-10", "source_label": "Exercise 1.6.10", "kind": "exercise", "statement": "Write a program in your favorite programming language to compute the greatest common divisor of two nonzero integers, using the approach of repeated division with remainders. Get your program to explicitly give the greatest common divisor as an integer linear combination of the given nonzero integers. Compare this method with the method of computing the prime factorizations of the two integers and taking the largest collection of prime factors common to the two factorizations in terms of computational efficiency.", "prelude": "", "natural_language": "Write a program to compute the GCD of two integers and compare it with the prime factorization method.", "requires": [], "references": [], "include_in_bench": false, "exclude_reason": "Programming task and discussion question"}
{"id": "goodman-1-6-ex-13", "source_label": "Exercise 1.6.13", "kind": "exercise", "statement": "(a) Develop an algorithm to compute gcd(n₁, n₂, ..., nₖ). (b) Develop a computer program to compute the greatest common divisor of any finite collection of nonzero integers.", "prelude": "", "natural_language": "(a) Develop an algorithm to compute the GCD of n₁, n₂, ..., nₖ. (b) Develop a computer program to compute the GCD of any finite collection of nonzero integers.", "requires": [], "references": [], "include_in_bench": false, "exclude_reason": "Algorithm and programming task"}
{"id": "goodman-1-6-ex-12-a", "source_label": "Exercise 1.6.12(a)", "kind": "exercise", "statement": "N/A", "prelude": "import Mathlib", "natural_language": "Whether relatively prime implies pairwise relatively prime for more than two integers.", "requires": [], "references": [], "include_in_bench": false, "exclude_reason": "Discussion/judgment question, not a fixed claim; and taken as an unconditional identity it is false in general (e.g. 6, 10, 15 are jointly coprime but not pairwise coprime)."}