Problem 1: Find integers a, b that do not have a greatest common divisor. Prove that the pair of numbers that you found are the only pair of integers...

Problem 1:

Find integers a, b that do not have a greatest common divisor. Prove that the pair of numbers that you found are the only pair of integers that do not have a gcd.

I do not know how to approach problem 1. Can you please help me prove this statement?

Problem 2:

directions: What is wrong with the following statements? Repair these statements and prove your revised versions.

(a) For all integers a, b, we have b|a iff a div b =a/b.

(b) For all integers a, b, we have b|a iff a mod b = 0.

I think the revision to 2(a) would be "For all integers a, b, we have b|a iff a div b =(a-r)/b

I think that 2(b) is incorrect because div and mod are defined for a,b Z,b>0.

I understand why the statements are incorrect, but need help proving them.

Q1)Let a= 3(1+√-5) and b= 3(1−√-5) in Z[√-5]From the point of view of Kummer and his "ideal numbers", verifying that 2 behaves like"the square of a prime" in the...