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Estou com uma grande lista de exercícios de PROPRIEDADES DO MDC (MÁXIMO DIVISOR COMUM), e não estou conseguindo entender quais os passos que tenho que seguir nas demonstrações, e gostaria muito de aprender este conteúdo, alguém me ajuda em uma questão para ver se eu consigo entender as outras?

Questão: Mostre que, se (a, b) = 1, a|c e b|c, então a · b|c

Added Translation from Portuguese

I have a large list of exercises PROPERTIES MDC (greatest common divisor), and I am not able to understand what steps you have to follow in the statements, and would love to learn this content, someone help me on a question to see if I I can understand the other?

question: Show that if $(a, b) = 1, a | c$ and $b | c$, then $a · b | c$

Andrés E. Caicedo
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benjamin_ee
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2 Answers2

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Proof Sketch:

Take the prime decomposition $c = p_1 \cdots p_n$.

Since $(a,b) = 1$, we know that $a$ and $b$ have no common factors. $(*)$

Since $a | c$, we know that $a$ is a product of $p_i$'s; similarly, $b|c$, so $b$ is a product of $p_i$'s.

By $(*)$ we know that $a$ and $b$ don't share any prime factors; thus, multiplying them together will give a product that divides $c$. "QED"

$$c = \prod p_i$$

$$a = \prod_{i \in A} p_{i}$$

$$b = \prod_{i \in B} p_{i}$$

$$(a,b) = 1 \implies A \cap B = \emptyset$$

$$a \cdot b = \prod_{i \in A \cup B} p_i \big| \prod p_i = c$$

Benjamin Dickman
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Another way is as follows. If $(a,b)=1$ then there are integers $\alpha, \beta$ such that $\alpha a + \beta b = 1$. Now $c = c\cdot 1 = c\alpha a + c \beta b$ But $c\alpha a$ is divisible by $ab$ (why?), as is $c \beta b$, so $c = c\alpha a + c\beta b$ is divisible by $ab$ too.

Matthew Towers
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  • Este Jeito E o Que eu Mais me interesso los entendre, na Verdade Curso Matemática, EO professora explica UO Mais Menos ASSIM, Que Só trabalhamos com UMA definição parecida, mas that TEM UM Sinal Negativo, Tipo αa - βb = 1 – benjamin_ee Jul 08 '13 at 17:25
  • Consegui...:) Obrigado pela ajuda – benjamin_ee Jul 08 '13 at 18:18