1.
i) if p is prime, a,b are integers, and p divdes ab, then p divides a or p divides b[Hint use result when proving the theorem there exists integers m, n such that g = m*a=n*b]
ii) If r is a rational number then there exist integers m,n such that n>0, r = m/n and m, n have no common factors.. [USE WELL ORDERING PRINCIPLE]
iii) PROVE SQRT(21) is irrational [Hint use part 1 and part 2]
For everything positive.
The consequence of the Well-Ordering principle is that any two positive integers has the greatest common divisor .*)
In that case we can write,
Where, .
*)Let .
Then by Zorn's lemma this set has a minimial element (this is greatest common divisor).
Thus, any two positive integers have a greatest common divisor.
I think that the well ordering principle referred here is the axiom “Any non-empty set of positive integers has a first term”. Then consider That set is not empty and has a first term N.
Now , so show that M & N cannot have a common divisor other than 1.