Advanced Math Proofs

1. For an integer a, what is the greatest common divisor of a − 1 and a + 1?

2. Show if a, b and c are integers with c|ab, then c| gcd(a, c) gcd(b, c)

3. Find all the integer solutions to each of the Diophantine equations below.
(a) 72x + 52y = 124
(b) 17x + 21y = 48
(c) −81x + 105y = −18

4. An oil company has a contract to deliver 250 000 liters of gasoline. Their tankers can carry 2500 liters and they can attach one trailer carrying 3000 liters to each tanker. All tankers and trailers must be completely full on this contract. Find the least number of tankers required to fulfill this contract. Each trailer, if used, must be pulled by a full tanker.

5.
(a) Let a = 15 and b = 210. Give the prime factorization of a, b, a² and b². Does a|b and a²|b² ?
(b) Prove the following statement: Let a, b ∈ Z, then a²|b² if and only if a|b.
(Use the prime factorization of a and b)

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