How to tell if something has a common factor

[rsala004]

if we have 6 and 9

and we break them down to sets of prime factors {2,3} , and {3}

if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

or in more specific to my interest...

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks

 

[mathman]

if we have 6 and 9

and we break them down to sets of prime factors {2,3} , and {3}

if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

or in more specific to my interest...

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks

Your main question has yes as an answer.

I don't know what you mean by

this only true for their sets of non-zero prime+composite factors

 

[Mensanator]

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks

By definition, a set of prime factors won't contain any composites.

 

[CRGreathouse]

I'm going to guess the question and then answer what I thought the question was.

Question: Given positive integers m and n, if the set M of primes dividing m and the set N of primes dividing n are disjoint (have an empty intersection), is m/n an irreducible fraction? [If this is false, is it at least the case that if the set M' of divisors of m and the set N' of divisors of n have {1} as their intersection, is m/n an irreducible fraction?]

Answer: Yes, if there are no primes in common than m/n is irreducible.

 

[ramsey2879]

If you are interested in the theory behind the gcd then you should check out these pages re Euclid's Algorithim and the Chinese remainder theorem.
http://www.cut-the-knot.org/blue/Euclid.shtml
http://www.cut-the-knot.org/blue/chinese.shtml