Can (2) Be Proved Without Knowing (1)?

In summary, the Greatest Common Divisor (GCD) is the largest positive integer that divides all given numbers evenly. It can be calculated using various methods and is important in mathematical applications such as simplifying fractions and solving linear Diophantine equations. The GCD is always a positive integer, but can be represented as a negative number in certain contexts. The GCD and Least Common Multiple (LCM) are related concepts, with the relationship between them being expressed as GCD * LCM = product of the numbers.
  • #1
annoymage
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0

Homework Statement



(1). If gcd(a,b) = d , gcd(a/d,b/d)=1

(2). If gcd(a,b) = d , then gcd(m,n) = 1 , where dm=a, dn=b
(i don't know wether this statement is correct)

Homework Equations



n/a

The Attempt at a Solution



i kow how to prove (1) and (2), but i only proof (2) using (1), my professor applying that (2) without us knowing what (1) is. seeming that they are other direct and obvious ways for proving (2),
so is there any? i can't see T_T
 
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  • #2
(1) and (2) says exactly the same thing. The same proof will work for both.
 

What is the definition of Greatest Common Divisor (GCD)?

The Greatest Common Divisor (GCD) of two or more numbers is the largest positive integer that divides all the numbers evenly.

How is the GCD calculated?

The GCD can be calculated using various methods such as the Euclidean algorithm, prime factorization, or listing all factors and finding the largest one.

Why is the GCD important?

The GCD is important in many mathematical applications, such as simplifying fractions, finding equivalent fractions, and solving linear Diophantine equations.

Can the GCD be negative?

No, the GCD is always a positive integer. However, it can be represented as a negative number in certain contexts, such as when using modular arithmetic.

What is the relationship between GCD and Least Common Multiple (LCM)?

The GCD and LCM are two related concepts in number theory. The GCD is the largest number that divides two or more numbers evenly, while the LCM is the smallest number that is a multiple of two or more numbers. The relationship between them can be expressed as GCD * LCM = product of the numbers.

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