Prove2Me

Coprimality with a product

Proved
WorkbookSource.problem_34519

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Suppose that gcd⁡(h,k)=1\gcd(h, k) = 1 . Show that for every a∈Na \in \mathbb N , we have that

[gcd⁡(a,h)=1  and  gcd⁡(a,k)=1]  ⟺  gcd⁡(a,hk)=1.[\gcd(a, h) = 1 \text{\ \ and \ } \gcd(a, k) = 1] \iff \gcd(a, hk) = 1.

Source: InternLM Lean-Workbook, record lean_workbook_34519 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.problem_34519 {a h k : ℕ} (h1 : Nat.Coprime h k) :  Nat.Coprime a h ∧ Nat.Coprime a k ↔ Nat.Coprime a (h * k)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_34519; Apache-2.0

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