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1. Let a ∈ Z and b ∈ N. Then there exist q ∈ Z and...

1. Let a ∈ Z and b ∈ N. Then there exist q ∈ Z and r ∈ Z with 0 ≤ r < b so that a = bq + r.

2. Let a ∈ Z and b ∈ N. If there exist q, q′ ∈ Z and r, r′ ∈ Z with 0 ≤ r, r′ < b so that a = bq + r = bq′ + r ′ , then q ′ = q and r = r ′ .

3. Let a, b ∈ N. The greatest common divisor of a and b exists.

4. Let a, b ∈ N. The set of common divisors is non-empty.

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