feat(Nat/GCD): Add dvd_gcd_mul_gcd_of_dvd_mul (#12246)

Mathlib/Data/Nat/GCD/Basic.lean

@@ -345,4 +345,23 @@ theorem Coprime.mul_add_mul_ne_mul {m n a b : ℕ} (cop : Coprime m n) (ha : a 
   rw [← mul_assoc, ← h, add_mul, add_mul, mul_comm _ n, ← mul_assoc, mul_comm y]
 #align nat.coprime.mul_add_mul_ne_mul Nat.Coprime.mul_add_mul_ne_mul
 
+variable {x n m : ℕ}
+
+theorem dvd_gcd_mul_iff_dvd_mul : x ∣ gcd x n * m ↔ x ∣ n * m := by
+  refine ⟨(·.trans <| mul_dvd_mul_right (x.gcd_dvd_right n) m), fun ⟨y, hy⟩ ↦ ?_⟩
+  rw [← gcd_mul_right, hy, gcd_mul_left]
+  exact dvd_mul_right x (gcd m y)
+
+theorem dvd_mul_gcd_iff_dvd_mul : x ∣ n * gcd x m ↔ x ∣ n * m := by
+  rw [mul_comm, dvd_gcd_mul_iff_dvd_mul, mul_comm]
+
+theorem dvd_gcd_mul_gcd_iff_dvd_mul : x ∣ gcd x n * gcd x m ↔ x ∣ n * m := by
+  rw [dvd_gcd_mul_iff_dvd_mul, dvd_mul_gcd_iff_dvd_mul]
+
+theorem gcd_mul_gcd_eq_iff_dvd_mul_of_coprime (hcop : Coprime n m) :
+    gcd x n * gcd x m = x ↔ x ∣ n * m := by
+  refine ⟨fun h ↦ ?_, (dvd_antisymm ?_ <| dvd_gcd_mul_gcd_iff_dvd_mul.mpr ·)⟩
+  refine h ▸ Nat.mul_dvd_mul ?_ ?_ <;> exact x.gcd_dvd_right _
+  refine (hcop.gcd_both x x).mul_dvd_of_dvd_of_dvd ?_ ?_ <;> exact x.gcd_dvd_left _
+
 end Nat