gcd_eq_one_of_gcd_mul_right_eq_one_right

English

There exist integers a,b such that gcd a b = 1 and m = a·gcd(a,b), n = b·gcd(a,b) when gcd(a,b) > 0.

Русский

Существуют целые a,b такие, что gcd(a,b) = 1 и m = a·gcd(a,b), n = b·gcd(a,b) при gcd(a,b) > 0.

LaTeX

$$$$\\exists a,b\\in\\mathbb{Z}, \\gcd(a,b) = 1 \\wedge m = a\\cdot \\gcd(a,b) \\wedge n = b\\cdot \\gcd(a,b).$$$$

Lean4

/-- If `gcd a (m * n) = 1`, then `gcd a n = 1`. -/
theorem gcd_eq_one_of_gcd_mul_right_eq_one_right {a : ℤ} {m n : ℕ} (h : a.gcd (m * n) = 1) : a.gcd n = 1 :=
  Nat.dvd_one.mp <| h ▸ gcd_dvd_gcd_mul_left_right a n m

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