Line in proof: $\gcd(a,b) \le \gcd(b,r)$ - why?

Question

Looking at the following GCD proof, I'm unsure I understand why line 4 and 8 are necessarily true. How is it that $\text{gcd}(a,b) \leq \text{gcd}(b,r)$?

$$\begin{align} &\text{gcd}(a,b) \setminus a \wedge \text{gcd}(a,b) \setminus b \tag{1}\\ &\text{gcd}(a,b) \setminus (a - qb) \tag{2}\\ &\text{gcd}(a,b) \setminus r \tag{3}\\ &\text{gcd}(a,b) \leq \text{gcd}(b,r) \tag{4}\\ \\ &\text{gcd}(b,r) \setminus b \wedge \text{gcd}(b,r) \setminus r \tag{5}\\ &\text{gcd}(b,r) \setminus (qb+r) \tag{6}\\ &\text{gcd}(b,r) \setminus a \tag{7}\\ &\text{gcd}(b,r) \leq \text{gcd}(a,b) \tag{8}\\ \\ &\text{gcd}(a,b) = \text{gcd}(b,r) \tag{9} \end{align}$$

Edit: Please do not edit the question to change $\setminus$ to |. Both are accepted notation, even if $\setminus$ is less common.

Beware that the answer you accepted is likely a circular proof in your context. — Bill Dubuque, Mar 23 '20 at 17:13

Maths Matador · Answer 1 · 2020-03-23T17:47:35.793

1

Here I am assuming that $a = qb + r$, where each are integers. For line (4), since $\text{gcd}(a, b)$ divides $b$ and $r$, then it divides $\text{gcd}(b,r)$. We can see this because any common divisor of $b$ and $r$ must also divide its greatest common divisor. So it follows that $\text{gcd}(a,b) \leq \text{gcd}(b,r)$. We can then apply the same argument to line (8).

Edit: As Bill in the comments noted, this proof in most contexts is circular and so should be avoided. Look at Bill's answer for better clarity.

edited Mar 23 '20 at 17:47

answered Mar 23 '20 at 15:42

Maths Matador

946

This is a circular proof in most contexts since it implicitly uses the gcd universal property $\ d\mid b,r\iff d\mid \gcd(b,r),,$ whose proof typically employs the above property. I explain what is intended in my answer. – Bill Dubuque Mar 23 '20 at 17:12
@BillDubuque duly noted, it slipped my mind that it implicitly used this property. I will write an edit acknowledging this. – Maths Matador Mar 23 '20 at 17:44

Bill Dubuque · Accepted Answer · 2020-05-02T17:33:57.553

The key idea of the proof is greatly clarified if we combine both directions. The first three lines of both directions combine to yield the following divisibility equivalence

$$ \color{#c00}d\mid a,b\iff \color{#c00}d\mid b,r,\ \ \rm therefore $$

$a,b\,$ & $\,b,r\,$ have same set $S$ of common divisors $\,\color{#c00}d,\,$ so same greatest common divisor $=\max S$

Remark $ $ We can simplify the equivalence by rewriting it as follows

$$\begin{align} {\rm if}\, \ d\mid b\ \ \ {\rm then}\,\ \ d\mid a\ &\iff d\mid a-qb\\[.2em] {\rm said}\,\bmod d\!:\ {\rm if}\ b\equiv 0\,\ {\rm then}\,\ a\equiv 0&\iff a-qb\equiv 0\end{align}\qquad\qquad$$

Line in proof: $\gcd(a,b) \le \gcd(b,r)$ - why?

2 Answers2