QuestionQuestion

1. For a, b ∈ Z with b > 0 define the set S = {. . . , a−2b, a−b, a, a+ b, a + 2b, . . .}. Give an argument for why S must have a non-negative element.

2. Are there integers a, b and c (all non-zero) such that a|bc but a/|b and a/|c?

3. Let a, b, c be integers. Prove the following statement: If ac divides bc and c ≠ 0, then a divides b.

4. Let a, b and c be integers. Prove that if a divides b and a does not divide c then a does not divide b + c.

5. Using the Euclidean Algorithm find the gcd(a, b) and write the gcd(a, b) in the form ax + by, where x, y ∈ Z (show your work!)
(a) a = 3953 and b = 944
(b) a = 4653 and b = 1324
(c) a = 5280, b = 3600
(d) a = 19200 and b = 3587

6. Let a, b and d be positive integers.
(a) Show that d gcd(a, b) divides both ad and bd
(b) Prove that gcd(ad, bd) = d gcd(a, b).

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