Trouble setting up to prove unique factorization

[lants]

This lemma the book states, I can't make sense of it.

Lemma: If a,b$\in$ $Z$ and b > 0, there exist q,r $\in$ $Z$ such that a = qb + r with 0 $\leq$ r < b.

Proof: Consider the set of all integers of the form a-xb with x $\in$ $Z$. This set includes positive elements. Let r = a - qb be the least nonnegative element in this set. We claim that 0$\leq$ r < b. If not, r = a - qb $\geq$ b and so 0$\leq$ a-(q+1)b<r, which contradicts the minimality of r.

Can someone help explain this to me? Why can't a = 3, and b = 8. Then no q,r could exist so that r is less than b, right?

 

[lants]

Is this considered a textbook style question? Sorry, how can I move it? I just saw number theory and posted

 

[Terandol]

What is wrong with $a=3$ and $b=8$? Here you simply take $q=0$ and $r=3$ (there is no condition that says either q or r have to be nonzero) to get $3=(0)(8) +3$ and $0\leq 3< 8$ as required.

 

[lants]

Wow I'm quitting now

thanks

 

[blue_raver22]

As a scientist, it is important to approach any problem or concept with a critical and analytical mindset. In this case, the lemma being referenced is a fundamental concept in number theory known as the division algorithm. This algorithm is used to prove the unique factorization of integers, which is a crucial concept in mathematics.

To address your concern about the specific example of a=3 and b=8, it is important to understand that the lemma is a general statement that applies to all integers a and b. In this case, the lemma is stating that for any given integers a and b, there exist two other integers q and r that satisfy the equation a=qb+r, where r is the remainder and must be less than b.

In the example you provided, it is true that a=3 and b=8 do not have a q and r that satisfy the equation. However, this does not invalidate the lemma. It simply means that the example does not fit the criteria of the lemma. In fact, the lemma is proven to be true for all integers a and b, regardless of specific numbers.

I would suggest reviewing the definition and proof of the division algorithm to gain a better understanding of this concept. It is also helpful to practice with different examples to solidify your understanding. As a scientist, it is important to continue questioning and seeking clarification in order to fully comprehend and apply concepts in your field of study.