QuestionQuestion

Prove the following:
1. Let a, b and c be integers. Prove if a|b and b|c then a|c.
2. if a|b and a|c then a|bx + cy for all integers x and y
3. Let a, b and c be integers with a > 1. Prove if a|b and a|c then a does not divide bc + 1.
4. Prove that every integer larger than 1 can be expressed as the product of prime numbers.
5. If a = qb + r then gcd(a, b) = gcd(b, r).
6. Show that gcd(a, b) = 1 if and only if there exist x, y, ∈ Z such that ax + by = 1. (Think GCD Characterization theorem)
7. Prove for integers a, b and c, that if a|bc and gcd(a, b) = 1 then a|c.
8. Prove if a, b, c are integers with gcd(a, b) = 1 and c|(a+b) then gcd(c, a) = 1 and gcd(c, b) = 1.

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