Theory of Numbers Properties of GCD (Greatest Common Divisor)

Question

I am a student in Undergraduate Mathematics, and I'm struggling to number theory ... I have this problem gcd, and do not know how to do it, and still do not study, congruences, Diophantine equations or, among other matters more advanced ... I'm used to divisibility, and some properties and/or theorems gcd ... Help me, please ...

question a) Show that if

$(a, b) = 1$, $\Longrightarrow$ $(a · c, b) = (c, b)$.

Note: Do not speak, nor write, english, use a translator, then I urge you to write without abbreviations. Thank you. — benjamin_ee, Jul 18 '13 at 18:52
I tried to get ... But not out ... Did ...
Demo: $(a⋅c,b)=j$ e $(c,b)=k$ $\Longrightarrow$ $j=k$ — benjamin_ee, Jul 18 '13 at 18:53
Have you already in your course the theorem that if $x$ divides $mn$, and $\gcd(x,n)=1$, then $x$ divides $m$? — André Nicolas, Jul 18 '13 at 18:54
Yes Thomas $(c,b)=d$; $d|c$; e $d|b$;
a

$d|c$ $\longrightarrow$ $d|a⋅c$ $\longrightarrow$

$(c,b)|(a⋅c,b)$ — benjamin_ee, Jul 18 '13 at 19:32

score 1 · Answer 1 · answered Jul 18 '13 at 19:16

1

Hint 1: It is sufficient to show: $$d\mid c \wedge d\mid b \Leftrightarrow d\mid ac \wedge d\mid b$$ for any $d\in\mathbb Z$ (why?)

Hint 2: For the "$\Leftarrow$"-implication, use that $d\mid ac$ and $d\mid bc$, so $d$ divides their gcd.

answered Jul 18 '13 at 19:16

Tomas

4,534

felasfa · Answer 2 · 2013-07-18T19:50:07.023

0

We use the following theorem from Number Theory.

$$ g=(a,b) \Longleftrightarrow ax+by=g $$ for some integers $x$ and $y$. **

Now consider $(a,b)=1$ then you can write $$ ax+by=1\quad (1) $$ Let $g=(a\cdot c,b)$ then invoking the above theorem again $$ ac(x_{1})+b(y_{1})=g\quad (2) $$ for some integers $x_{1}$ and $y_{1}$.

Now if we show the following $$ c(ax_{2})+b(y_{2})=g $$ we are done. To do that, look the hint below.

Hint Consider the two equations above. Multiply equation $1$ by g and so some algebra.

edited Jul 18 '13 at 19:50

answered Jul 18 '13 at 19:42

felasfa

1,632

I assume that "NT" is an abbreviation for "Number Theory" – apnorton Jul 18 '13 at 19:47
yes edited to reflect your comment. – felasfa Jul 18 '13 at 19:50
The above theorem is known as Bezout's lemma. It's proof involves the concept of the Euclidean division/algorithm. This might be trivial, but exceeds divisibility properties, which were required by the OP. – Tomas Jul 18 '13 at 19:53

Theory of Numbers Properties of GCD (Greatest Common Divisor)

2 Answers2