Bunuel wrote:
If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?
Given: \(x=10m\) and \(y=10n\), for some positive integers \(m\) and \(n\).
(1) x = 4y. If \(x=40\) and \(y=10\), then the answer is YES but \(x=80\) and \(y=20\), then the answer is NO. Not sufficient.
(2) x - 3y = 10 --> \(10m-3*(10n)=10\) --> \(m-3n=1\) --> \(m=3n+1\). \(m\) and \(3n\) are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So, \(m\) and \(3n\) do not share any common factors but 1, which means that \(m\) and \(n\) also do not share any common factors but 1, therefore the greatest common factor of \(x=10m\) and \(y=10n\) is 10. Sufficient.
Answer: B.
Hope it's clear.
We can also note on statement 2 that since X and Y are both multiples of 10 and the difference is a multiple of 10 then 10 must be their GCF, if x and y had a higher GCF say like 20,30,40 etc... then the difference of two multiples of 20 will never yield 10 as a difference, same with others
Just my 2cents
Additional note: If the second statement were x-y = 10 we could also know that the GCF cannot exceed 10 because the GCF is always less than the difference between two numbers and exactly 10 when these two numbers are consecutive integers. Therefore we would know that these two numbers x and y are consecutive integers.
But that's for another problem I guess
Hope it helps
Gimme some freaking Kudos!!
J