If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y?
A. can not be determined
B. 1
C. 2
D. 5
E. 6
[spoiler]this was extracted from the BTG problem set; the explanation suggests an error and trial method by plugging first 1 and 1 for x and y, then 3 and 2 for x and y. I have opted to use the prime factorization for x and y instead; given x and y have the greatest common factor (GCF) =1, any primes added to their products would still make GCF=1 unless we are not certain about the numbers x and y having both the factors 2 and 3. Hence for us it's difficult to determine from 2x and 3y which numbers have factors 2,3 and/or both => therefore it's answer A. Such problem could include many multipliers not only 2 and 3, and the problem's solution would need to be analytical rather one suggested 'trial and error'. What do you think?[/spoiler]
A. can not be determined
B. 1
C. 2
D. 5
E. 6
[spoiler]this was extracted from the BTG problem set; the explanation suggests an error and trial method by plugging first 1 and 1 for x and y, then 3 and 2 for x and y. I have opted to use the prime factorization for x and y instead; given x and y have the greatest common factor (GCF) =1, any primes added to their products would still make GCF=1 unless we are not certain about the numbers x and y having both the factors 2 and 3. Hence for us it's difficult to determine from 2x and 3y which numbers have factors 2,3 and/or both => therefore it's answer A. Such problem could include many multipliers not only 2 and 3, and the problem's solution would need to be analytical rather one suggested 'trial and error'. What do you think?[/spoiler]
